Am I missing something?

Yes. The fundamental role of the Arpeggiator and a basic understanding how they work... which can lead to discovering a method of how *you* are going to use the feature. Other than that... But it is a topic worthy of discussion. You are correct, there are far too many Arp Phrases to just listen to them all. Here’s some things to think about...

If you want/need a musical phrase to play a specific way, you would be best served using the sequencer to play it back, or use the convert type “Fixed”...either is capable of playing exactly what you played (provided you do not exceed the limits of Arp-making).

In general, a useful note-oriented Arpeggio phrase might be one that you can use to adjust to the chord quality and Key that you use to control it. These are created with the ORG NOTE convert type.
“Fixed“ and “Org Note” are the same except the latter adjusts to the Key you are playing in.

Understanding the difference between a Arp Phrase and a Seq Pattern Phrase, is key. Perhaps you are better served by using the sequencer to playback your phrase rather than using the Arpeggiator, at all.

There are merely 10,231 Arps...(slightly less than a bajillion)... if it ever crosses your mind to look for a specific phrase among these, you are perfectly correct in avoiding spending your time listening to 10,231 Arp Phrases. Don’t do that!!! That is not recommended. If you “know” what you want - play it.

Amazing, how many folks miss this fundamental. All arp phrases were ultimately played by some one. The one you’re thinking of IS NOT THERE! (In all likelihood)
A “useful” note-oriented Arp is one that can adjust to your control input — major, minor, diminished, augmented, dominant...

Rather, look to the Arp Phrases for inspiration. Instead of looking for something specific when auditioning Arp Phrases, I would say it is going to be IMPOSSIBLE to find what you want. There are only 10,231...an actual bajillion would be needed to cover every possibility.

I think it is far better to dial up an Arp Phrase and be inspired by what you can do with it. If you find yourself auditioning phrase after phrase after phrase... STOP.

If you start with a specific (original) idea, okay great. Record it. The first skill in Arpeggio making, is learning what makes a good (universally useful) arpeggio. Not all musical phrases make good arpeggios. Analyze what you’ve played - determine if it would make a good Arp Phrase. And determine HOW you would like it to adjust to your input.

You outline a specific example Phrase that you want the Arp to play... not enough info to determine much without the rhythmic component. Rhythm is a key in Arp making. What do you need it to adjust to in terms of chord quality/Key.
If it is just going to repeat that over and over, that’s easy = Fixed Note.
If it just going to play that phrases over and over but change Keys, that’s easy too = Org Note
If you want it to adjust chord quality, then you need to consider the Original Root Note. If you make the Root C, the phrase will always start on the 5th. And if you flat the 3rd it will play in a minor key.

I recommend you study some of the 10,231 Factory Arps... to get a better understanding of what an Arp Phrase is, study how it is created to be useful in most musical situations. See the several tutorial in the LEARN section of this site.

For example, a chordal rhythm guitar arpeggio, when initially recorded, might be either played with three or four note chords. Say we use four notes and we play a C Major 7 chord C-E-G-B... it is the rhythm of the chords that is most important. To be musically useful to the person attempting to use it as an Arpeggio Phrase in their song, they want an interesting rhythmic pattern... because they are going to supply the chord qualities. Adjustments to the 3rd, 5th, 7th are recognized and create the different chord qualities.

If you are attempting to record musical melodic phrases (as opposed to chordal parts) well, I question your selection of the Arpeggiator as the tool you want to use. Now, not that you couldn’t, but try listening to 50 or 60 Factory Arp Phrases... just as an idea of what makes a useful Arp phrase.

First, you’ll discover very few melodic phrases because they don’t translate well as useful musically. What that means is the person looking for an Arp to use, would come across a melodic phrase and not know what it’s used for. What you’ll find is rhythmic presentations of typically one chord quality. You begin to understand how this is useful in a generic sense. It ‘suggests‘ but leaves room for the player to apply it.

Even two different chords is generally NOT DONE (I’m not saying never done, just less useful). If you record a II7-V7 progression, it is not as useful, in a generic sense, as an Arp Phrase that is just the first chord.

If you convert Dminor7-G7 as an Arp Phrase, the person using it cannot define *when* the chords move.
What if you play them two beats each, and the person looking for an Arp wants them for four beats each... or one beat each.

So what does make a good Arp?
Your phrase... find a factory phrase that is similar
Work with it - see what you can and you cannot get it to do.
And spend at least a week or more with Arp’s before changing how they currently work. I think when you find out what they can do, (they do really well) and when you discover those things they cannot do (which is like play melodies).

All that said, you can do whatever you want with Arps — you only need to make universally useful Arps if you plan on marketing them yourself... if you do, you become much more conscious of their general usefulness, (and less concerned about a specific use case). Making specific use case Arps (ones that will only work for a specific song or situation) is what it is...

but, when I hear somebody lament over the “bajillion“ Arp Phrases... I know that you’re never going to find the exact one you need... I have to mention, that you’re correct. STOP LOOKING You are unlikely to find a specific phrase. And no description is going to define what you want — trying to describe what you want in an Arp phrase is like dancing about architecture (to borrow from Frank Zappa). You are going to have to hear it, then apply it, and likely adjust it. If I say “R&B rhythm guitar”... or “strummed Bossa nova“... there are still compelling reasons for you to actually listen to it and audition it in context. No word, category or description is going to tell as much as listening to it — it is, in fact, the only way to know. Because ONLY YOU know what your looking for...

IF YOU KNOW WHAT YOU WANT: Play it!
If you have a general idea, find an Arp that is close - that inspires - then alter it.

Arp making is a skill that can only be appreciated once you start to recognize just how difficult they are to make.
How are you going to use it once it is an Arp?
Then the big question... does it need to be an Arp?
Perhaps it maybe better suited for playback from the Sequencer.

Extra Credit
Often when composing, I’ll use a Drum Arp for the basic groove, a Bass Arp - the simpler the better just to outline the root, I may use a rhythm guitar Arp to setup a feel, then make a Performance and record a chord progression... basically outlining the framework of the composition. I later go back, redo the drums, add a proper bass line, I use the Arps to inspire.
I did a tutorial with examples
“Smooth it Over“ - a 4-Part Motif XF Performance
I use it to show how you can use a Performance and Arp Phrases to construct musical sections.
I posted them on Soundmondo