Chelsea should be able to put trust in every coach they employ, like Ten Hag at Manchester United and Arsenal with Artata. Not all coaches are able to do a great job in just one first season, even if it is not impossible, it will be really difficult because of the adaptation process.
Pochettino is a coach who can be categorized as a classier coach compared to many other current Premier League team coaches. It's a shame that Chelsea fired him just because he failed in the first season, it seems like they don't appreciate his work in one season. I think Chelsea will be just as bad if this habit continues, they won't be as successful as before.
Every time a coach changes there will definitely be a change in game plan. Pochettino and other coaches will never have the same style of play and strategic decisions. This will make the players adapt again to a new style of play, different portions of training, and even have to adapt to several new players that the coach has brought into the team. It's unreasonable to expect a season of great success instantly, it's very difficult when the team is really bad.
Chelsea has appointed the 5th coach in a space of two seasons, Chelsea does not trust any coaches they have appointed for the two seasons because they believe it is possible to have a successful season with their current squad, the management has not been patient with the four coaches they fired in between two seasons, no one has been there for two seasons, in their first season without winning any trophy they got fired. I was surprised to see that Pochettino was been sacked either because I don't consider him as a failed manager to Chelsea, since Tuchel left the club he's been the first manager to build the players and secure their chances to play in the next Conference League.
Pochettino has come so far with these players, they have already adapted to each other, the management thinks because Enzo Maresca became successful at Leicester City this season, they think he can manage Chelsea next season. The new coach has to adapt to the players for his first season to have a better year otherwise the management might sack him too if he didn't have a great season as they wanted.
I agree with you, Chelsea might continue to have a bad season if they don't trust any of their coach for like two seasons, I believe if Ten Hags was Chelsea's manager, he could have been sacked since last season, we cannot even remember his existence at the club because they do not have a patient like Manchester United.