He realized there was zero chance of the Cards dealing him and he was smart enough to know that trying to force it in this short window by going nuclear was not going to change that and would just hurt himself (trade value, salary potential, HOF potential) as much as the org.
The thing is though, MB needs more than him to squash the "I want out" stuff. He needs to him to be dedicated to the org, be a leader, be all-in and not just on game day. This is where I think MB told him that him to give it a few more games (if not the rest of the season) and if things don't get better and the players are still in mutiny mode that he will force changes, and I'm sure PP said he'll go with the flow and hold him to it in the offseason.
What those changes are I don't know, and it likely depends on the time scale of how they are going to measure improvement and success. It could be as small as forcing the defense scheme to change a bit more towards what the players are comfortable and successful with. It could something bigger, like promising a new DC. Or, it could be as far as promising a new HC next year. It sounds crazy, but if the Wilks and staff have lost PP, there is a chance (not saying likely, but a chance) he's lost most/all of the team. At that point you either let the inmates run the asylum or get a new HC, or else you'll get uninspired play out of these guys.
I'm not sure if Wilks has lost all of the players yet, but I can see for guys like PP and Fitz it is very tough on them. They have performed at the top of their game, made lots of money, locks for HOF (Fitz 1st ballot, but PP will likely get there)... all they want is ring, and the first step is to be just good enough to get to the playoffs. This team is isn't light years from what it was just a season ago, so I'm sure in their minds their isn't even any light at the end of the tunnel to give keep that dream alive. Fitz is way too much of a pro to ever ask for a trade, but if I were in their shoes I can't say I wouldn't be thinking about the prospect of going to a team that isn't a sh*tshow, let alone a contender.