Out of curiosity, what's in Part 1? In other words, what sound did you start with that is occupying Part 1 (or 1-2, or 1-3). This would have determined your system/master effects.

You can see this by, from the [PERFORMANCE] (HOME) screen, touching the "Type/Name" box to the left of your Parts to switch it to "Name" and then read off the names of all Parts.

It is likely you are inheriting programming from previous Parts… or are adversely affected by current Performance Common settings… so without more information it will be almost impossible to say exactly why, or what is happening in your case.

But since you mention “I am inserting all 3 parts of the Rock Hard organ, one at a time, as 1 at a time seems to be the only way to do it.”….. this clue indicates that since it seems the “only way”, (it is NOT) perhaps you are using an incorrect way for what you want to accomplish… or are simply not aware of what programming already exists and is interfering with your goal.

When creating a Performance, and ‘searching’ the catalog for new sounds, you can add new Parts to existing Parts using:
__Performance Merge — this method allows you to place all 3 Parts into your new creation simultaneously if you wish (so it is obvious you are not using “Performance Merge”

__Part Category Search — as the name implies, you can only select one Part at a time. And while you can certainly insert a 3-Part sound into your creation one Part at a time, this clue tells us that is indeed what you have done (or are doing).

When you use the [CATEGORY] function to build your Performance and you are using “Part Category Search” you are replacing existing Parts with your selection.

This is where you can ‘inherit’ existing settings or you can opt to bring along the Part programming of the selection from its previous HOME.

When using “Part Category Search” you will see options for “Parameters with Part”
There are three (or four) options:
Mixing, Arp/MS, Scene, (Zone, if applicable)

When these options are green you are set to bring along the selected Part’s previous Part programming:
“Mixing” includes Volumes, Pan, Note Limits, etc
“Arp/MS” includes the Arpeggios previously assigned and Motion Sequences
“Scene” includes any offsets from the previous location
“Zone” only appears if Zone Master is active—allows you to bring in the settings for the MIDI Zone

Green indicates you want to bring the selected Part’s previous programming. Why these are useful is — say you are programming your Performance to be a keyboard split, electric bass/organ; you set the bass C-2 thru B2, the organ C3 and above;
the first organ you tried needs to be replaced, you want to be able to bring in a new organ, but you want that new selection to respect your C3 split point… this is where you opt out (grey) the Mixing box… the new organ will replace the previous one, but respect your split point… as well as other programming you may have done.

Leaving the option green would likely bring the new organ into your new creation mapped across all keys… thus forcing you to redo the Note Limits.

Suggestion:
Learn how to review how your Control Assignments are affected when you build a new Performance from existing programming.
“Performance Merge” is available to use when you are adding “+” a Part, or multiple Parts, to a blank Part slot. You only see the “+” icon in an empty Part slot. Touching the “+” launches “Performance Merge” and lets you set the Source to an individual Part # or “All”
If, however, the slot is occupied when go to Category Search, then you are looking to *replace* the existing Part (and “the only way” is one Part at a time).

HINT: You can DELETE a Part completely by tapping the Part in the screen; a pop-in menu appears: select “DELETE”.

Need to know: notice I carefully mention Part Control Assignments are brought along to your new creation. This is to differentiate them from the Common Control Assignments.

There can be only one set of Performance Common settings per Performance… so no matter how you bring in new Parts they will never bring along their upper Common/Audio settings — this means things like Reverb Effect, Variation Effect, Master Effect, Master EQ, nor any of the Common Knob and Super Knob programming.

There can be only one set of these, and they belong to current HOME location. You must review and then redo any Common Assign Knob functions (the newly relocated Part is now likely in a new Part # slot… you will find that actual programming still remains… waitin for you to decide if you would like to activate it in this new Performance.

For example, the organ you are bringing in likely occupied Part slots 1, 2, and 3 in its HOME location, and now you are placing it in Parts 4, 5, and 6. Once you ‘tell’ your new program where to find their target, you can activate the Common Assign and Super Knob programming (assuming you have not exceeded the 16 Assignments per Knob).

I can be more clear, if you can fill in some of the info. Let us know.
See solution to similar problem in a recent thread:
Some Organ patches are silent

BTW, I couldn't reproduce the same issue. I had to guess what the rest of the Performance looks like. Since harmonica is the first instrument in the tune that I thought a keyboardist would cover, I started with "FM Harmonica". Interestingly, I didn't find the sampled harmonica to be as close to that overdriven blues harmonica sound as the FM sound. I didn't take very long trying to find alternatives. At any rate, Part 2 I added "Tenor Growl AF1" because the second sound in the tune was a sax. This instrument/Part doesn't really impact anything. Then I added "Bright Trumpet AF1" to Part 3.
Again, this doesn't impact the organ. Then I added "X Draw Organ" by pressing the [+] on Part 4. I turned off keyboard control on Parts 1&2&3 and left them on for Parts 4, 5, and 6 (the organ). Pressed [PERFORMANCE] (HOME) to select no Parts and then struck piano keys. The organ worked fine as I think it did for you.

Then I deleted Parts 4, 5, and 6. Then I again pressed Part 4's [+] to merge in a different organ and selected "Rock Hard". This added the different organ to Parts 4-6. Playing this organ I could hear everything - there wasn't any issue.

This just means I followed a different path than you did. I can't really reproduce without knowing at least your Part 1.

... and, this is no part of the problem - but to me the organ sounds more like a transistor organ than tonewheel (including Hammond B3). There's a "VX Surf Organ" (as in Vox) or the "FM YC Combo Organ" that sounds more like the recording than any tonewheel-ish organ does. Both of those are 1-Part sounds and multi parts may be contributing to your issue depending on how you're building your Performances.

One more comment:

The "Rock Hard" organ's 3 Parts all have dry level set to 0. This means all sound is coming from the system effects (Variation and Reverb). If the system effect returns are set to 0 and the global Variation+Reverb knobs are spun to "0" (full counter clockwise) then this would cause the issue seen (no sound yet the meters are showing levels).

You could fix by changing the system effect return values or by increasing the variation+reverb sends.

The problem with "Rock Hard" in combining with other sounds is that "Rock Hard" has its distortion coming from the system effects. Therefore, it is not very "compatible" with system effects of many other instruments unless they will sound OK either fully system-effect dry (no reverb+variation sends, dry level cranked up in the Part) or sound OK with the distortion.

"Rock Hard" will sound a lot different without the system effect's distortion. This may be the case if you inherit system effects from another Performance (whichever was the first Performance recalled that you built the Performance off of - typically labeled in Part 1's "Name" ).

When combining different sounds - rather than digging into all of the settings and trying to resolve the conflicts - it may be easier, at first, to simply change the order you add sounds together. Starting with "Rock Hard" then adding parts from 4+ for your other sounds would change the order and would place Rock Hard with its correct effects and programming and then the other sounds would inherit.

Experimenting with the order like this (choosing something else to be the first loaded) may fix problems. You may not know "why" - but that can be learned at a slower pace over time.

The "Rock Hard" organ's 3 Parts all have dry level set to 0. This means all sound is coming from the system effects (Variation and Reverb).

By contrast the 3 parts of "X Draw Organ" all have dry level set to 127.

OP - you can see the dry level settings on the 'Part Settings -> General' tab when you edit a part and view the 'common' (not element) settings.

One of the nice things that Yamaha does is dedicate a pair of effects to each part, so those sounds will still have those effects if combined with other sounds. This is in contrast to, for example, the Korg and Kurzweil boards which, on one had, give you the flexibility to put as many of the available total effects as you want on a single sound, but then can no longer assure you that the sounds will be the same (i.e. those effects will still be there) when combined with other sounds. But it's worth noting that, even so, it is still possible for a Yamaha sound to not sound the same when it is used in combination with another, if its sound is dependent on the system effects rather than (or in addition to) the individual part effects.

Thanks All for your great and detailed responses - there's a bit to go through so I'll get back to you and let you know how I went.

@Jason - thanks for your comments on the actual sounds - I was also thinking that the sounds under the Sub-Category of Combo were more like the original - I was going to use Vx Full Bars but the sound, at least in my practice space sounds "small" (not sure what this corresponds to technically - stereo width or reverb/delay?) so I had decided on Rock Hard. I may change that again based on your comments.
re the Harmonica - I am using Bluz Distort and find that I can only get the really distorted, overdriven sound if I play 2 notes at a time. I tried the FM Harmonica with a distortion effect and the same thing seems to be the case - not sure what I would need to do to make the distortion come through for single notes?
Re the Tenor Growl AF1 - yes I am using that on some other performances and our guitarist actually helped me fit it into the rock band sound by getting me to extend the delay and a bit more reverb.
I actually started the performance with an electric bass as I was previously doing the bass duty, and I had also added a rhythm part for practice purposes, but after all this I might just bite the bullet and create a whole new performance.

Thanks Bad Mister,

Some additional info - I didn't mention that while I am trying to replace parts 4,5,and 6, that I have actually filled up the part 7 and 8 slots.

I tried the "Performance merge" approach (touching the "+" under part 4 after I had deleted parts 4, 5 and 6) and under Source, the only options I could get were Part1 - part 16. The "All" option did not appear.

I thought I would test this by creating another performance starting with a single part bass (Precision Flat Wound) and then using the "+" on the vacant part 2 slot. This procedure did actually allow (or default to) ALL under the "Source" box.

It seems that performance merge can only come into play when the slots to the right of the selected part slot are all empty?

Does this make any sense in terms of the architecture of the system?

The problem is in the system effects as I mentioned before. I didn't know exactly what you had - but I theorized that perhaps the Variation Return at the common/audio level would be set to 0 causing this issue.

Now that I know you started with "Prec Flat Wound" - I can reproduce your issue.

1) Used [CATEGORY SEARCH] to bring up the "Prec Flat Wound" Performance. Not merging - just bringing up that single Performance by itself.

2) Added "Rock Hard" (organ) to parts 2, 3, and 4 by pressing Part 2's [+] button and selecting "Rock Hard"

3) Muted Part 1 (bass) and played piano keys. The organ level meters were dancing but no sound was coming out.

A Fix (one possibility):

a) Touch the Performance name
b) Choose "Edit" from popup -- you'll be editing the Common/Audio Section
c) Select the "Effects" -> "Routing" menu
d) Note that "Var Return" is set to 0. Set this to a higher value like 64 or 127. Now you can hear the organ.

The problem is that "Prec Flat Wound" sets up the variation+reverb returns as 0. And this is what I previously mentioned would cause this problem. Set them to something other than 0 (really, just variation is sufficient) and that will fix the "no audio" issue.

As mentioned before, "Rock Hard" depends on 100% of its audio to come from the system effects (system effects = variation + reverb effect buses). You could have also altered this by editing the organ Parts and changing "Dry level" to something other than 0 (64 through 127 for example). That's another possible fix - make the organ not rely so heavily on system effects by increasing the dry level.

The system effects still may not be correct for "Rock Hard" (the sound will be different than the original organ sound). But optimizing this would involve tradeoffs and deeper diving than I think you need to do at the moment.

Kick the tires on just increasing the variation return value (detailed in a-d above) and see how that suits you.