I think Thomas Tuchel was sacked because at some point he wasn't getting wind for the Chelsea team which his boss wasn't comfortable with and having in mind of replacing him with a better coach everything became worse for the Chelsea team. I believe Tochel wasn't sacked because of emotion rather he was sacked because much was expected of him which he couldn't deliver and after all, none of the coaches brought back the lost glory to the Chelsea team
Tuchel had bad relationship with Chelsea director and owner during the last summer. He disagreed with transfers in the club and it continued after the season start. Some bad results during such bad time are enough for Chelsea owner to take advantage of it to sack Tuchel. It could be a decision which comes from individual conflict rather than for the club benefit. After Tuchel, Chelsea made two bad choices with Potter and Lampard and they consequently had very bad results even spending hundreds of $M in this season.
now that Pochettino has been employed, it is expected that he will take the team back to the first team where they belong I just hope that he will be given more time and not be sacked after few games.
Pochettino is a good coach and had good experience with Premier League. He built up Tottenham with limited resources and transfer budget, even helped Tottenham to play in a Champions League final. A few seasons later, Pochettino did not break out more so it can be his limit or Chelsea will be a club where he can win titles, Premier League and Champions League.