Rodgers had some unkind things to say and Rodgers action on the field was wrong. If a HC is going to take the responsibility for wins and loses then run the damn play I call and according to Rodgers he did not like the play calling and would change it. If that is the case then those loses are on Rodgers. He is the QB not the HC
Mike McCarthy just spent a season "studying up" after being the head coach of the Packers for 13 seasons.
He didn't "study up" because he had nothing better to do while being paid $6-8M by GB in 2019.
He was informed by the rest of the NFL at the end of last season (there were 8 openings) he needed to engage in some some self-reflection in order to be considered for another head coaching opportunity.
As he now at least tacitly concedes, Mac studied up because he needed to get better. Good for him. Whether it worked remains to be seen.
That Rodgers, and practically everyone else associated with the Packers, said some "unkind" things about Mac at the end of his tenure is more due to how Mac was performing (or not performing) his job than anything else.