Kompany when he was manager of Burnley in the Championship was totally different from the Burnley in the Premier League. You forgot to add that he was the person to bring them out of the championship to EPL with a marvellous performance. You can't compare that very team to the other teams in the EPL, so I really don't agree with you on the narrative of coaching a relegation team then going on to win the league the next season with another team. Kompany would have still struggled if it was Wolfsburg he was in charge of and not Bayern Munich.
Burnley is different from Bayern Munich and as such, he will still struggle as a manager if he had managed a team like Wolfsburg in the Bundesliga as you have said. He did better since he had a better team and a better management. Good management results to good play from the team and that is what happened when he took over the charge of Bayern Munich.
For a manager like him that was just starting out, he needed the job to show what he got even if the management can’t provide all of his needs, so his best form wouldn’t be really known managing a smaller team with no good welfarism. Now that he is coaching Bayern Munich, he has everything to show how good he is and that is why he is excelling here now. Giving him another team with no good welfarism will still make him to end up not been the best of manager in the competition once again.