If I was in the market for a lathe that small, I’d probably just buy a slightly larger mill and work on rotating parts vertically, TBH.
If you were committed to wanting a small lathe and CNC, I’d just put linear rails on some of the larger 8020 profiles and rig a spindle from a cartridge spindle. Honestly, though, an 8x16 lathe is kinda bare minimum for steel parts unless you are working VERY slowly. The 7x14 I had worked, but at that scale, I’d be working hard to implement a clough42 ELS conversion so that threading didn’t suck so bad.
If this is genuinely something you want advice/help with, feel free to email me.
EDIT: I don't really have room for a manual mill which would be large enough for this sort of thing to make the parts I want(the Grizzly would be really pushing its limits).
It's more idle speculation at this point in time --- I did save the electric motor from replacing the garbage disposal recently (cracked housing) and I also have a key duplicator I fished out of a dumpster (working!) and really, it should be workable to fabricate something given the tools I have.
If you did a writeup on making a lathe w/ 8020 profiles and a cartridge spindle (source?) I'd be interested --- what would the total landed cost be?
I think there is plenty of room for innovation in hobby machines, but IME, hobbiest value work envelope maybe even more than pros.