A big problem for Putin, Shoigu and Gerasimov is that elite units like the 4th Guards Tank Division and the 2nd Guards Motorized Rifle Division which were given the honor of the Moscow garrison to defend the capital were absolutely mauled in Putin’s invasion of Ukraine.
They spent months recovering in Belarus and are still in the Ukraine theatre supposedly in the Luhansk area in very poor shape.
Anyway, it is not clear these units would not side with Prigozhin given the massive losses they sustained under the incompetence of Shoigu and Gerasimov.
So Putin has to rely on air power and the Rosgvardia to defend the capital. And Wagner has their own air defense systems.
For Prigozhin to go this far implies he is confident of internal support inside the Kremlin...or he’s full of hubris and is recklessly deluded. Either way its a huge gamble for him.