You're spot on - the notion that we improved after it was announced Pellegrino was to be replaced by Pep in the close season is simply bollocks. It was Pellers himself who revealed Pep's impending arrival in a press conference on 1 February 2016. At that point, we had 44 points from 23 games, three behind leaders Leicester (and with a better goal difference).
We then took 22 points from the last 15 games, a drop from over 1.9 points per game before the announcement to below 1.5 afterwards. We finished in fourth spot, 15 points off the top, and scraped CL qualification only on goal difference.
We'd already qualified for the League Cup final, which we won on penalties, and we squeezed past one of PSG's weaker modern incarnations to make the CL semi, only to put in a supine display against Real where we didn't even seem to try to win the tie. Neither of these things changes the reality of our dreadful league form once it was known Pellegrini was on the way out.