Is my proof of Unitary matrices preserving length legitimate?

I've been learning about Quantum computing, and central to the idea of a quantum logic gate is that gates can be represented as Unitary matrices, because they preserve length.

I couldn't get an intuition for why U^(†)U = I would mean that len(Uv) = len(v).

After a lot of messing around I came up with these kind-of proofs for why this would be the case algebraically.

https://samnot.es/quantum/unitary-matrices/

Is anyone able to validate/critique these proofs?

I'm not clear on how these map back to the more formal notation proofs for the length-preserving property of Unitary matrices.

Does anyone have any more visual way of grasping why they preserve length?

Thanks!