What if Prigozhin backing down is just a juke, and he's planning something under the table?
Looking back at some evidence, I believe that one of the reasons why Prigozhen folded is because he failed to get a truly good hand. The push to Moscow wasn't THAT quick and enough troops were transferred to Moscow (including by airlift) to stop a few thousand fighters rushing in.
Another important factor, not counting some very minor guys, no one supported Prigozhin. Yes, many were reluctant to obey orders (oh, I can only imagine all that sweet internal purging in the Russian military that is going right now) but it is not the same as joining.
Nevertheless, he had a good negotiating position to receive some guarantees.
1) His guys captured the supply and C&C Rostov hub, jeopardizing the war effort on the Donbas front and having resources to resist for a long time.
2) He captured several military airfields, threatening to destroy (or even use against Russia) substantial assets.
3) While taking Moscow (key buildings) was out of the question, the mere event of fighting there would be extremely damaging.
4) Ukrainian counter-offensive wasn't going anywhere. A few more days of chaos could cause a collapse on the frontlines
Lukashenka seems like the only true winner of this event, he got a personal army. They may have dubious loyalty but they are FAR better than Belarusian armed forces that can do nothing against Russia or even possible Belarusian uprising supported by, at the very least, Poland and Ukraine (and it will be supported, trust me)