I don't think Pochettino's stay at Chelsea has been something to say the club have improved because if you say he should stay I think this guy will only handle Chelsea like he did with Tottenham without any trophy and you'd agree with me that the fans and equally the board don't have that time to wait, we know how desperate they are especially the fans.
People can change, from players to coaches, and Pochettino with his talent can change and upgrade himself, and win trophies. If coaches like Ranieri, Spaletti can win trophies after many years and seasons as a runner up, without league trophy, don't feel so pessimistic with Pochettino.
He is a good head coach and I believe if he manages a big club, is given good budget for player transfers, chances to build up a squad he wants, and has enough time and patience from the club, he will succeed, sooner or later, matter of time only.
I knew that Pochettino did not win big trophies so far in his coach career but I don't underestimate his capability as a head coach.
I'm not trying to be pessimistic here is just the reality that I'm saying about the man Pochettino, those coach you mentioned won trophies after a long while because of how they think, their mindset changed after getting very close to winning a title that's why they went back and did some calculations made some changes where they made lots of errors and they got it right by winning some trophies but for Pochettino he's like a stagnant water, no changes still on the same losing box and you expect him to receive an applaud from the fans? No it can't happen.
I don't see him having that fighting spirit in any way so I can't say he's trying (but he should do better) and he should stay with the club, time is running out and he's not the type that can outstand pressure in any game especially from smaller teams. If you talk about time and patience, I don't think the fans have that, all they want is a positive results not mind how but get it done.