I’m not even sure Taiwan makes that claim, but if they do then I’m fine with that.

They do. The official government line currently is that they have no need to formally declare independence (which might trigger a Chinese invasion) because they already are an independent country by most meaningful measures (which is true of course)

I had heard they view themselves as the legitimate government of a (unified) China.

That used to be the official position decades ago. But apart from a few old nationalistic farts maybe, nobody on Taiwan really holds that position anymore.