The private part is still important. It's a legitimate fleet of military aircraft that would rival the air forces of many small countries that is directed by a man who really isn't accountable to us outside our borders.
This isn't new territory, it's just that you have to go back a few hundred years to see small groups of private individuals accumulate this much physical force. Just think about some of the disagreements that have occurred over scare resources for responding to the pandemic. Imagine how that picture can change.
This just doesn't make much sense as a concern.
There would be little point in training against them otherwise.
Notably, Erik Prince of Blackwater/Academi notoriety played fast and loose with weapons import/export/ownership rules, and came very close to being criminally prosecuted for it. Retired Admiral William 'Fox' Fallon resigned in disgust when he realized how far outside the rules Erik Prince was operating.
When Erik Prince suggested letting his private firm handle security in Afghanistan as an outsourced contractor, then SecDef (General) Jim Mattis was dead against it, and blocked it all the way.
This was one of the big concerns if the US/NATO ever went toe to toe with the Soviet Union. With the way modern weapons systems work if they both went all at it, it would not be long before one side or the other had enough of an edge up - just through attrition - that there would only be two options left to the losing side - surrender or go nuclear. You can't have two fully capable super powers fighting one another all out for years any more. You'll run out of stuff that works too fast.