Are you saving these as "Fixed" or "Normal" Arps? Ref Manual page 16 has the details

Determining how Song/Pattern data is converted to an Arpeggio – Convert Type
MIDI sequence data (of Song Tracks) can be converted to Arpeggio data in one of three ways, according
to the Convert types below. These types can also be selected independently for each destination Track—
providing enormous flexibility and performance control.

Normal (Normal Arpeggio) - The Arpeggio is played back using only the played note and its octave notes.
Fixed - Playing any note(s) will trigger the same MIDI sequence data.
OrgNotes (Original Notes) - Basically same as “Fixed” with the exception that the Arpeggio playback notes
differ according to the played chord.

My problem is if I hit a single note when recording the MIDI data, that's all I get when using my new Custom User ARP as I play various chords on the piano.

If I were to guess, I think you have converted this as a "Fixed" arpeggio, and that is why it plays the same note irrespective of which chord you hold.

I would record 1 measure of Scene 1, then three measures of Scene 2 so there's some "white space" before it loops around again. My problem is if I hit a single note when recording the MIDI data, that's all I get when using my new Custom User ARP as I play various chords on the piano. If I record a C Major chord, again, my User ARP only sounds good when I play major chords. The same would be true when "gluing together" four factory Bass ARPs that work great when used individually, but are no longer intelligent when I assemble them via MIDI data record.

Can these type of intelligent ARPs only be made by Yamaha, or is there a way a lay-person can accomplish something similar?

You can make chord intelligent Arps but...

There is an old joke about drummers hanging out with musicians. This is because the drummers have their own rules (they march to their own beat). When you record a Drum Arp, you want the exact notes that you play to playback. This is called a “Fixed” note Arp conversion.

The drummer doesn’t change the notes in response to the chord changes. The kick drum, snare drum, hihats, toms, and cymbals remain the same regardless of the Key or chord quality.

A musical Arp Phrase is going to use the Arp Convert Type = OrgNote. (Original Note) This signifies that you will need to designate a Note to represent the Note that when fed to the Arpeggiator will cause the Phrase to play at its original pitches.

Say you record in C Major 7 and you want this Arp Phrase to be assigned to your Brass horn riff... you would record the Phrase you want to sound when C Major 7 is input. You need not only to feed the phrase information, you would input the Note that would determine where on the Keyboard this Phrase will sound (OrgNote). If you put in C3, then when C3 is played your Phrase will play its original pitch.

If you create an Arp with this data as the information... when you voice a C minor 7, the phrase will dutifully play your Phrase but it will flat the third, and flat the seventh to convert the phrase to the chord quality of your controlling input. If your Control information includes C3 the Phrase will play right there in the same range as your original Phrase, if your control information includes C4 it will play the phrase as a Cminor 7 an octave higher than the original notes... and so on.

If you trigger an F7, the Arp will raise your Arp Phrase a musical fourth, it will flat the seventh. Basically it takes the CMajor 7 data and makes it play F7... to do that it must either raise the info a 4th and flat the 7, or drop down a 5th and flat the 7 to make it an F... and since the third is already major, it only needs to flat the 7.

If you notice, listening to the factory Arp Phrases as examples— they all outline one chord. This is an important point.
This allows *you*, the player, to determine what chord quality they play at any time with your control input. They do not consist of multiple chord changes.

Arps are designed to play one chord quality — this allows the player to input the chord changes wherever/whenever they are necessary.

If you are recording a Brass line as a Phrase... you must determine where to designate the “OrgNote”
If the musical line is recorded in a specific Key, set the root Key and general area of keyboard you wish to be the control area.

Set the convert type = OrgNote.
Select the appropriate OrgNote
Be sure to name your Arp data - finding it later will depend on your naming system.