The quote from Duffy indicates that the SpaceX team doesn't have any real knowledge about or experience with the FAA systems. Seems like they're being brought in just because they're a Musk company.
Political pettiness, and speculation about whether Musk got some special favor or advantage that would not be available to other companies aside, it doesn't seem like a bad idea to get cross-pollination and share ideas with other organizations and fields. SpaceX may not know much about ATC but they probably do know something about monitoring and control and collision avoidance in rockets and satellites.
What I wrote stands though, the article seems poorly sourced and incorrect about its interpretation of the tweet.
It's been reported that 400 new hires on probationary status were let go. None of them were involved in safety related operations. The FAA has around 45,000 total employees, so it's a rather tiny percentage, in any case.
Procurement bids should be transparent and avoid the illusion of conflict. This is the complete opposite of that. It's hard to take Musk's campaign against "fraud and waste" serious when he's awarding the contracts to himself.
I think this move is a mistake, but what your describing there is just competent management.