It's interesting that its specifically Wagner who make the claim.
Certainly. Prigozhin has been criticising the Russian high command. The high command has therefore sent Wagner into a meat-grinder battle to teach its owner a lesson and make him less cocky. The continuing failure of Wagner to achieve much on the battlefield embarrasses and rebukes Prigozhin by showing Wagner can't do any better than the Russian army. So now it is Prigozhin embarrassed and frustrated, trying to raise public enthusiasm for every building his outfit manages to grind control over.
There's been a lot of speculation about all the various russian oligarch and regional leader getting they're own PMC/militia and what that could mean if Russia started to disintegrate into a bunch of pseudo state.
I think this is about long-term war, where Russia plans to send a lot of people out to die. Mercenaries serve two functions: they keep deaths off the Russian military's books, and also make it less complicated to bring in foreign troops (who may not be desired in the Russian army proper). After that, they just want to spread over several PMCs so there's no private army individually capable of threatening the security of the state.
Personally, I think there are a lot of arseholes like Prigozhin who fancy themselves great leaders with macho, militaristic fantasies. Of course these guys are going to love private armies. It feels almost like going back to the feudal levy concept, where nobles, city councils and so on would raise regiments for national service (which sort of survived in adapted form to the 18th / early 19th century).