You can imagine the shame when gimli put it on and became an elf

The ring worked on the Dark Lord differently, so it definitely might be race bound

Yup, all we know is that there's some connection to the shadow world ("and in the darkness bind them") and other rings of power, we don't know the specific impact it would have on each individual or race.

Hobbits don't desire invisibility, they're just good at not being noticed.