I know plenty of Burnley fans, and not one of them wanted Dyche to be sacked at the time. Yes there were definitely Burnley fans who thought he had reached the end of the road and it was a time for a change, but I think they were clearly in the minority, an increasing minority but still a minority. Then again, quite a few fans from what I gather thought that he would leave the club at the end of the season regardless of whether they stayed up or not, especially with growing friction between the owners and him. Most Burnley fans I heard on 5 live and Talksport disagreed with the decision.
I think it was a different story compared to Stoke sacking Pulis in 2013 or Brighton sacking Hughton in 2019 for example, where large numbers of fans, while appreciating the excellent work that they had both done, completely agreed with those decisions. Adjusted for inflation, Pulis had a noticeably bigger budget at Stoke thanks to the backing of Peter Coates and Bet365, than Dyche has had at Burnley (millionaires in a league of billionaires).
I remember when Huddersfield sacked Lee Clark in 2012, there was widespread outrage from fans of other clubs, but many Huddersfield fans (including most of the ones that I spoke or listened to) completely agreed with it.