As much as I dislike Allegri and his stupid views on football, I don't think letting him go now will solve our problems. I don't even trust in a sudden increase of the performance level - unless players start actively boycotting him (which could happen down the line).
The team has chronic problems, which have been going for years now, which range from poor squad building, poor mentality of the coach, mentally fragile players (individually), poor fitness, poor focus, excess of injuries, injury prone players, club dettachment from supporters.
You don't solve all that at once.
They should consider doing it after the WC, if reaching the 4th place is becoming more and more unlikely. But only for a short fix. Do not sign anyone for a long contract, under any circunstances.
Next season, first step is signing Giuntoli to oversee all decisions. Absolute priority. Get rid of Arrivabene. Let Giuntoli decide, along with Agnelli, Cherubini and Nedved, who could be the guy to lead the team.
I'm not sure how to get rid of Cherubini and Nedved though, and it's not clear if they are indeed the most important part of the problem.