The coach will be the first person to blame when a team fails to perform well, but we also have to see how Chelsea management always makes hasty decisions regarding their coach. The reason is that they don't give the coach much time, in fact they only give the coach one season and then they fire him.
Many teams fail in their first season with their coach, but in their second season many finally make good progress and now Chelsea is not doing that.
Pochettino, for example, had to be fired at the end of this season, even though in terms of his game I think he could show progress, Chelsea were not satisfied so they fired him. It is very difficult for a coach if he is only given a very short time.
The management of a club could also play a role to the failings and failures of any club. ( A good example would be Napoli last season as the management made poor decisions that didn't help matters. Secondly, I'd say the owners of Manchester United the glazers who didn't care about the club but making profits.
Also, the management refusal to back a manager properly can also lead to stuffs like this also. The coach is also included and I completely agree with you on that.
Chelsea is a club who's got no patience for any coach, and lately coaches that are brought in are not given the freedom to do what they want. It's sad to see that they'll always want to influence the decisions of thr coach. Chelsea FC is a good example of what I'm saying.
Pochetino was beginning to do well and the players fighting and backing for him. I just don't know the discussion he had with Todd Boehly, so it's what it is.