I understand that the intuition behind a unitary operator is that it preserves the length of the vector it acts upon. Also $U^\dagger U = I$. Doesn't that just mean that $U$ is just an invertible operator which preserves distance? And that $U^\dagger$ is the inverse?
If not, where am I making my mistake in reasoning? How should I rewire my thinking to avoid this flawed paradigm?
If so, does that then mean that all distance preserving and invertible operators are also normal operators?