Multiple things can be true at once. Wilder may not have been considered for the Burnley job. Wilder may never have approached Burnley. But there clearly was connection between Wilder and it seems highly likely that Wilder was expecting to move to Burnley at the first opportunity.
Looking at the timeline and history, Wilder had strong connections with the American investors who took over Burnley and it seems highly likely that he thought that when Dyche moved on that the job would be his. It is also possible that the American investors really didn't rate Wilder as highly as thought they did and, also, that they had zero interest in paying a fee to get him from us. I think it's most likely that Wilder was played by them and burnt his bridges with us expecting Burnley to make an official approach.
We know that the one thing that Gibson won't tolerate is disloyalty. There is clearly a difference between agreeing to let your manager talk with a prospective new club; a whole other thing to find that your employee has been trying to leave behind your back. Gibson and Wilder had a "clear the air" dinner but I suspect that behind all the handshakes and smiles, they both had their fingers crossed. Wilder pledged loyalty but proceeded to ensure that his representatives kept their contacts out for the first chance to jump ship. Gibson pledged support but then used Bausor as the "no" man. The relationship became toxic with both sides looking for an excuse to end it and blame the other one.
It wasn't just Burnley. The relationship with Scott was clearly difficult; Wilder absolutely did not want to be here for the medium term and develop players. He wanted promotion and he wanted it now. He was only prepared to pay lip-service to the new sustainable model. The January window was poor and also coincided with key players struggling with form and fitness, possibly from the after-effects of Covid. Put all this together but mask it with a fantastic cup run and you get to see a classic example of something that looked fine on the surface but was rotting from the inside.
Probably Gibson should have let him go at the end of the season but at that point what Gibson probably needed to decide was whether Wilder was the problem or the solution. Gibson could have gone all-in on Wilder; sacked Scott and bought the old warhorses he wanted to fire us to promotion. Thing is, he had tried that with Warnock and (to a lesser extent) Pulis and it hadn't worked. However, at that point, Scott hadn't shown much to indicate that he could steer the club and there may have been no compelling candidates so Gibson either kicked the can down the road or decided to give the twin-track approach one last go. It didn't work but, possibly more through luck than judgement, we may have found a coach and team who will work well with Scott.