I think we may be overinvesting in what is essentially a Potempkin village or performance piece.
Prigohzin knew, starting out, that at the end of the specific day that his troops would be strong enough to fight, but too weak to stay because an army starved of resources eventually dies. An army on the march without air supiorty eventually dies. An army without fuel eventually dies. No matter where he turned, if Putin prioritized attacking him, his entire force would be bogged down within a week, and destroyed within three. Supplies matter. Yes, Prigohzin has anti-aircraft, but how many liters of fuel can an anti-aircraft section carry? Not enough to avoid the main contingent of his army being stuck in a fixed position when the fuel runs out and the artillery starts raining down on them, so the only determiants would only be time and how much damage becomes self-inflicted by both sides.
Putin also knows this. His only problem is he stares into the mirror so often, and refuses to listen to others. Typical Idi Amin complex. This was a test of clout, not between Prigohzin and Putin, but between Prigohzin and the army colonels opposing him and their factions. Maybe Putin has become so isolated that a coup attempt was the only thing that would fix his mind on something else. If that something is a familiar face, he has the smarts to work things out in his head. No mysterious threat, just the rattling of a cage to get his attention.
Prigohzin shouted "Hey!". Putin shouted "Hey!" back. End of argument.