Not sure if you've ever flown on a private jet but you just drive up to it, there is no TSA or security measures in place, you literally take your car onto the tarmac and walk onto a jet, it's marvelous.
With that said I don't know how many of these congressmen/women travel via mass commuter air transit (RFK died in a private jet, Sen. Ted Stevens died in a private plane, etc.) but I'm willing to bet a large majority of them don't, and therefore don't get to encounter the standard procedure or are ushered through unabated. Imagine some geek in a blue shirt telling John McCain he has to take off his shoes and belt to come though the check point... very unlikely.
There's a bit of a false equivalency between JFK Junior's private jet (RFK was assassinated by Sirhan Sirhan) and Ted Stevens' death aboard an Alaska bush plane. Small propeller planes are not an uncommon way to travel in Alaska since much of the state is still wilderness.
On a less important note:
"you literally take your car onto the tarmac and walk onto a jet, it's marvelous"
Wait, do you just leave your car parked at the tarmac for the whole trip?
I'd imagine most people able to afford a private jet (or to let a friend borrow it or whatever) can similarly afford some sort of valet service. That, or maybe there's some sort of parking space right by (or even inside?) the hangar for the private jet where you could leave the car. I'm a bit curious now too as to standard protocol on it.
My guess is that is the only way the problem gets taken care of quickly.