In my view, Chelsea is not a team that has patience and Chelsea is a team that cannot respect the process. Because anyway, if Chelsea can respect the process and give a second chance, then surely Pochettino can still do his best for his second season later. Indeed, Pochettino failed but anyway, in the end Pochettino also managed to make better results for Chelsea. So yes, even though Chelsea will appoint a new coach for next season but anyway, if for example the new coach cannot provide good results for Chelsea later in his first season. Then of course, Chelsea will do the same thing will sack the coach, because sacking the coach at Chelsea seems to have become their standard when the team fails to finish in the top four.
What you say is also true that Chelsea don't have patience so they always give short time to the coaches they have. It doesn't just happen this season, but in previous seasons they did the same thing. For example, Thomas Tuchel, where they didn't give Tuchel a chance and immediately fired him, and after that they even changed coaches at least twice after firing Thomas Tuchel.
And now that they are doing the same thing, I'm afraid that if they continue like this it won't get any better for them. because coaches also need time to show that they can handle a club. We can see from successful teams with their coaches, sometimes it takes years for them to become solid and create excellent chemistry between the players and Chelsea can't wait for that process.
They should be able to learn from what they experienced before, they can evaluate at least to give the coach time, especially if the coach shows good progress.