My honest opinion, Gattuso should quit management and become... a pundit on TV.
Valencia with Lim are complete disaster. No idea who makes the sporting decisions there, complete lack of vision as they keep appointing clowns as managers.
Don't think Gattuso is that bad. Had some bad luck and was probably mostly sacked because he wanted to get some new players.
Gary Neville was the last Valencia coach I actually blamed to fail. Managers after him did their best in tough conditions and then you have Marcelino who overperformed but he is one of the best Spanish coaches for me so not really surprising.
Gattuso was doing well initially tbh, but things went bad later and they are in relegation fight.
Honestly I would have preferred if he stayed, but I can't blame them for firing him.
This is what happens when you are stuck with an owner(s) who either doesn't care, or is incompetent, or simply has other ideas and priorities different from those of the members and the fans.