I'm not sure why everyone's being so philosophical, the ring works just fine on Frodo.
How so? Let's recall what the ring does by reading the User's Manual, conveniently printed on the inside of the ring itself:
One ring to rule them all, one ring to find them,
One ring to bring them all, and in the darkness bind them
SoWell,
- Bring them all: Frodo is certainly found by/finds/happens upon a whole bunch of ring bearers - most of the human rings and all of the Elven ones. As for the Dwarf rings, well, that feature was broken to begin with and didn't work for Mairon either. So, [✔].
- Find them: When Frodo puts the ring on, I recall he is very noticeable to other rings bearers (at least, the Human rings), and vice versa. See also this detailed answer. Now, probably a more experienced/corrupted/powerful bearer of the one ring would be better with this perceptive skill, but I'll put this one down as [✔].
- Bind them: The Elven rings avoided the binding by design, and the Dwarven rings just didn't have that effect like Mairon wanted to, so it's basically the Human rings. Given that 'to rule them' is a separate point than the 'binding', you could claim that they're still bound to the ring, and in many senses to whoever it is that's wearing it, even if he's not ruling them. So, [sort-of-✔].
- Rule them: This is the point about which everybody's fussing. Now, ok, maybe you don't get to boss (non-)people around that easily, and maybe you need to be a powerful Maia for that to actually work. So, [✖].
Three out of Four is pretty good, considering how Frodo is not really the ruling type. It's not clear why anyone should expect the ring to do other things for him.