Milan is a club that doesn't change coaches very quickly. Because the Milan management knows very well that any coach needs time to adapt to the players in his team. It is foolish to think that the performance of the team will improve immediately with the change of coach. Moreover, most of the times we see that the performance of the teams becomes unstable after the change of coach. This is why the Milan management is not interested in a quick coach change.
Stefano Pioli coached Milan for the last 5 seasons. I wouldn't say that Milan's performance with Pioli was poor. But I look forward to seeing if the players can adapt to the likes of Paulo Fonseca. Paulo Fonseca coached Roma and Lille in the last few years. With Paulo Fonseca, Roma and lille's performance was strong, no way to say that.
Milan perform under Stefano Pioli not really poor because success winning scudetto last two season, but small fund allocated for transfer top players make Milan difficult reach higher level perform but keep consistent finish on Champion League spot standings every season. After sacking Stefano Pioli, Milan still not announce yet who candidate become their manager for next season and seems waiting after Euro 2024 or waiting official releasing from management appointing new manager for next season.
Right now have many top manager as free agent after sacking by their team before from Thomas Tuchel, Xavi Hernandez and Juventus head coach former Massimo Allegri could be candidate list become Milan manager for next season because he was Milan former last several season before.
I agree with you that Milan's performance with Stefano Pioli was not too bad. However, the Milan team cannot dominate Serie A. But Stefano Pioli can't be blamed for that. And the Milan management isn't blaming Stefano Pioli either. Milan management is aware of their squad's weaknesses. They are well aware that the Milan squad is not strong enough. There are several weaknesses within the squad. If you want to strengthen the squad, you have to spend a lot of money. Otherwise, Milan will not be able to dominate serie a.
The three coaches you mentioned are all talented and experienced. Tuchel, Xavi and Allagri are all three very good players. I think Milan team will benefit if one of these three is given the responsibility of coach. Allagri has been in Serie A for a long time. He is well aware of the Serie A teams.