Though I think this could have been done in one writeup, I like the idea.

The origin, it seems to me, is accompanying, or accompaniment.

Of course, not limited to jazz, but any time--in my case--a keyboard, piano, or synthesizer is used to accompany, another instrument or instruments, or a voice.

Even, and especially, when I work up a comp for myself when I sing, say Satin Doll by Duke Ellington, As Tears Go By by the Rolling Stones, or my own stuff.

I'm teaching a young student of mine this now. She has the most large voice, and will be overwhelming when she can do her own comping