They were totally legit variants of V and I, especially in cursive. Cursive U/V was much closer to U than to V.
When cut in stone, V was easier to cut, and stone tends to be preserved better, so we see V more often. But at the same time, people would write U when writing.
What happened in mediaeval times is that people decided that U and V are two different letters. But both U and V existed long before that, they just were two forms of the same letter.
And since our lowercase letters are based on cursive forms, it makes much more sense to use u in the lowercase. Which is what modern sources that don't distinguish U and V do (e.g. Oxford Latin dictionary would not use "U" in uppercase and "v" in lowercase).
> Back when Lutetia was called Lutetia, it was written with V.
Well, it was also written with E and not e, and T and not t, but you seem to have no problem with that.
Lvtetia makes absolutely no sense. Lvᴛᴇᴛɪᴀ might be OK, it's at least consistent (albeit hard to type). But Lvtetia is a Frankenstein's monster: you freeze the evolution of V into u, but keep evolutions of other letters intact.