I didn't know much about Wagner or Prigozhin before this stunt. However, the bizarre nature of it made my thinker itch, so I spent a few days studying up. Initially I thought Putin and Prigozhin had cooked up a 'pre-coup' that would enable Prigozhin to take the reins from Putin when he retires, which I expect he will do before he gets assassinated (as always comes to dictators that overstay their rule after their edge begins to dull). Putin's not a fool, and knows his edge is dulling. He's well beyond his prime, and that lots of hungry younger competitors seek his seat.

But that didn't really make sense either, because the time to pass the crown was while Prigozhin had the momentum from the growing support he'd generated, and that didn't happen. I realized that Prigozhin has no interest in ruling Russia. He only wants to rule Wagner. I realized this only when I learned Shoygu had determined to forcibly incorporate Wagner into the MoD.

Putin and Prigozhin are bosom buddies. Putin gave Prigozhin Wagner safely out of the reach of Shoygu by this ruse. There are clearly other candidates for the Presidency than Prigozhin, some who want it enough to spend years in Russian prison for it, like Navalny. I don't know if Shoygu has realized how close he came to getting terminated with a sledgehammer, but when the VDV and other forces refused to tackle Wagner and let them march on Moscow, he must have pissed himself with fear.

He got enough of the Wagner veterans to spread their expertise across the MoD forces, which they desperately needed, but didn't get to take Wagner out, which Shoygu clearly wanted to do. Putin was confident in Prigozhin not having any interest in the throne, but Shoygu had no such trust, and Wagner's rapid growth to ~50k elite troops not only made him very nervous politically, it showed up his own lack of capability to create such an elite force very, very blatantly and often.

Putin sacrificed some face to save Wagner from Shoygu for his buddy Prigozhin, but has more than enough to not be significantly risking political comeuppance. Anyone else that thinks they can stand up to Putin will quite quickly learn they aren't Prigozhin. He'll still have to deal with succession later, but probably not too much later, and he can count on Prigozhin till the day he dies.

Thanks!

Edit: there's been a lot of speculation and talk about Putin losing face, but he's got ~90% favorable rating from Russia. Shoygu, not so much. The failure of the MoD personnel to war on Wagner, and worse, some of the MoD joining Wagner's march on Moscow, is a vote of no confidence in Shoygu, not Putin. He's hanging by a thread right now, and he's got a myriad of subordinates considering a go at him.