I'm more wondering whether Prigozhin will assist Putin into an ill-advised high altitude endeavour than vice versa.
No matter what face he puts on things, Putin has to be very seriously damaged by the Ukraine failure, and there are surely Russian officials seriously thinking about a post-Putin Russia. Putin might therefore want options from outside the Russian state to balance threats within - hence Prigozhin. But that also hands Prigozhin a lot of power to dethrone Putin.
That's the trick, isn't it? Authoritarians climb the ladder to power by exploiting division and using power plays, but at the same time, fear the same shit happening to them, so they use division to try to avoid letting anyone else getting too much power. Which is how you get crazy shit like duplication of resources and lack of cooperation between different people who should by all rights be cooperating(The Luftwaffe during WW2 had it's own ground troops because Goering didn't want to give any resources to the army and the Japanese Army and Navy during WW2 fucking hated each other to the point they were effectively fighting different wars).
Putin doesn't want the head of MOD to off him but at the same time probably doesn't want Wagner doing the same thing, so he has to walk that tightrope between letting anyone have too much power. OTOH, this also means that the military and Wagner are divided during an actual war and it hampers their effectiveness on the battlefield because people aren't cooperating to fight Ukraine(and likely fighting for glory and resources against each other).
Apparently after Solader Wagner and MOD both claimed credit for the victory and downplayed that the other was even involved and now Wagner has been cut off from recruiting prisoners so there's some bad blood to be sure.
Allegedly there was some incident in Syria where the US was gonna bomb some area and called up Russia to ask if they had any troops there. Russia said no and the US dropped bombs. What Russia didn't mention was Wagner had troops in the area who got killed that the MOD conveniently didn't mention. Then again, Wagner isn't technically part of the Russian military so....yeah.