Don't you still have to go through TSA even to board your own private plane? I could be mistaken...
Technically speaking, TSA, who are not federal agents, have to be invited onto an aircraft by the Captain.
They're a federal agency, but not of law enforcement (AFAIK).
As I understand it, they can be comprised of actual law enforcement personnel, including federal. But don't necessarily. And even if a VIPR team included federal law enforcement personnel, it's not clear that they're necessarily acting with federal law enforcement authority.
As if the TSA wasn't a confusing-enough bureaucratic monstrosity.
The old rules are simply not something I'd be confident applies. In the trivial case, sure, a gate agent isn't a federal officer. But if a TSA agent were to appear at a private aircraft, that would be out of the ordinary. And once you start talking about out-of-the-ordinary set-ups, it's worth noting that even a random TSA gate rapist may have out-of-the-ordinary legal authority in some situations. To say nothing of what law enforcement officers may be operating under the TSA's purview in said out-of-the-ordinary set-ups.
I know an attorney who flies their own jet, and if the TSA "agents" are doing something "out-of-the-ordinary", that is the time when there is a "badge check".
I think this is a moot point, as what VIPR teams are typically for securing public transportation or public events, neither of which is a private jet. Air Marshals, although actual federal agents, don't typically fly on private jets unless under Space-A (invited) privileges.
As to what VIPR teams do, a small airport is a VIP away from all sorts of federal officers asking all sorts of questions, private planes or not.