You think he might become a figurehead pulled by Prigozhin?
Well, it would certainly be more palatable for the rest of the world than dealing with Priz.
@AmeliaWarnerBros as @Igotjelly says no one really knows at this point.
I am not a trained observer, this are just my thoughts
If Prigozhin gets in power he'll have to work with the administration of government, as everyone does. At the moment Wagner is lean, very mean and keen. When they start actually governing, that leanness will soften and their keenness will slow. You can't get computer technicians to work at keeping the network going at the point of a gun for more than a few hours; you can't force doctors and nurses to work for long the same way; taxi drivers are still going to take an hour to get from A to B in the rush hour. Forms take time to be processed and that slows things up.
So if he gets in power, he'll have to learn how to rule once he's destroyed his enemy factions. He will have to rely on the existing civil service and on Elvira Nabiullina, the Governor of the Bank of Russia for economic matters. That will slow him up a bit.
Civil society within Russia is going to become less safe in the medium term either way. All those veterans with PTSD are going to be volatile at best and violent at worst. The children of such men are going to be brutalized and will become brutal in their turn, unless great efforts are made to curb this which just isn't going to happen. If Priz is in charge, Wagner soldiers might be able to swagger around and make it worse, faster, but I'm not sure it'll be that much worse in the long term.
It seems possible and even likely that Russia moves its society onto a permanently more militaristic footing.
Priz will have to honour some of Russia's contracts, specially with China - the Govt signed a contract recently where China can gain access to Russia's rare earths more easily. If he doesn't honour them, it will become a rogue state that no one at all wants to do business with.
They will, as a country, retain their seat on the UN but they're going to be pretty untrusted for a long time to come.
In the end time will pass and Russia will be more militaristic, but it will also stabilize and then slowly links will probably be built up again - merchants and tradespeople will always try to make money. The oligarchs are unlikely to go away and they want wider contact with the world.
The West has been Warned In No Uncertain Terms that they need to build up their militaries again. This doesn't necessarily mean war as such, it means simply that war is less likely to happen because Russia is opportunistic and will prey on anyone it considers weak. Some of the -istans might have to worry much more than the West.
In the end, there is a small chance a wider and worse war will break out but for the short term I think it's much more likely that Russia's going to be concerned with itself more than anything else. Looking further than the short term, the cost will be huge to Russia in terms of civilization and people killed but in the medium to long term things will settle.
Regarding longer term stability I think that's going to be a problem because of food shortages. Combined with the ecological changes coming, it's not a good scenario. A more militaristic Russia is probably not going to want to work on managing the changing world climate with others constructively.