Not surprised at all. I - and many, many people I know - have worked with numerous American companies. And when they acquire a business, the first thing they do without fail, is fire people. That's what they do. And this is completely on brand.
Having said that it was time for Dyche to go. All other teams around us making managerial changes too. And let's be honest - Dyche didn't "save" Everton from relegation. He was just lucky there 3 teams worse than us...
Not sure what the answer is going forward, but I would take much more pride getting relegated with a fighting and combative team in the final season at Goodison than getting relegated with shotless dross we are subjected too.