This isn't strictly hypothetical for me.
In Shroompunk, humans are all planetouched, either genasi or the equivalent of aasimar/tiefling/feytouched/aberrant. This is only partially hereditary and does not break cleanly on ethnic or cultural lines.
We're counting this as One. Henceforth, all of my nonhuman kith will be expressed as combinations of two D&D races.
Kappa: Dragonborn + Tortle
Tinker: Warforged + Kender, but malicious
Konga: Grommam + Bugbear
Vanara: Vanara + WoW Goblin
The rest of them contain one of the forbidden races (Myconid + Dwarf; Kobold + Halfling; Dromite + Halfling; Elf + Satyr), but nine kith is also a lot more when they're pulling double-duty as full classes.
For something more familiar, I'd probably ignore your "No Elf/Dwarf" rule, cut the half-races (including halfling) and the dragonborn and tiefling in favor of the orc, lizardfolk, thri-kreen, goblin, and gith.
Dragonborn could be a lizardfolk subrace, and make planetouched a whole "universal subrace" subsystem in the Manual of the Planes.