Construction of the complement of universal Turing machine - where is the catch?
Let V be a Turing Machine constructed the same way as U but with the final states switched


This construction won't work. This is a pattern in automata theory, by the way: for deterministic deciders, flipping final states provides a new proof for closure against complement. For all others, it usually fails.

Here, observe that your semi-decider $U$ of L is, by definition, of the form:


  Accept after finite time if $w \in L$; loop otherwise.


With your construction for $V$, you get:


  Reject after finite time if $w \in L$; loop otherwise.


This is not a semi-decider for $\overline{L}$!

Intuitively, you can not flip a non-termination; you don't get the result you would flip!