Prigozhin was being politically displaced by the Ministry of Defense. His only hope was to displace the Ministry of Defense right back. Putin and the rest of the military didn’t bite, and Prigozhin’s position was strategically untenable. Maybe he could have taken Moscow, but everyone important had already left, and he couldn’t hold Moscow once he had it. So he accepted exile over a fight to his certain death, because he’s a mercenary and mercenaries don’t die for a cause.