Skin effect is very well present but not at audio frequencies, its totally irrelevant to talk about it here.
In a good design, no way a change in primary mains side could effect audio frequencies in signal path or amplifier. If it happens then there is something drastically wrong with the system. The primary side has very little current requirement, any decent wire will suffice, its trivial.
The secondary side is well buffered with clean DC is the norm. The amplifiers these days are excellent in performance too making further improvements in the system very hard and of diminishing return.
As DIYers one must find and fix real problems, not imaginary ones. Many times we find imaginary problems, attempt to fix them and 'find improvements'. Well, we fixed something so there has to be a positive change. This is called confirmation bias.
Experiment in audio is very tricky and misleading to the person doing it. Only objective measurements can reveal the differences. Once measurements are proven and in the right direction, should we do subjective evaluation. That subjective evaluation has to be independent and blind ie no bias.
And the fact that we tweak things and we enjoy it, is a different matter altogether.
