Using the terms "GA" or "private planes" is a hugely broad brush because it's not just private jets. There is absolutely GA low-hanging fruit that, in my opinion, should be grounded first in the event of a shutdown. For example there's a shitload of Cessna 172 and Cherokee student pilots clogging up the system, requesting VFR flight following and a Class Bravo clearance for their flight from Denton to Waco. Not to mention their departure tower is already handling 4 touch-n-go's in the pattern (half of which want a short approach) and 5 in line waiting for takeoff. That should be the first to get cut and would help alleviate both tower and enroute ATC: no pattern work at a towered field and VFR on your own. Another would be instrument students who want to knock out 4 practice approaches on a single IFR flight plan (each with the published miss please!) at Conroe, Hooks, and then Sugarland. Those take up far more ATC bandwidth and patience than a Gulfstream on a standard departure/arrival whose going to get in/out of the area quickly.
After that you could work your way up through grounding piston, then turboprop, etc. using either engine type or MTOW, maybe even adding a Part 91 vs. 135 element to it. Point being if a government shutdown forces us to start grounding aircraft, there are many different ways to split up the GA world that would have a meaningful impact on alleviating ATC's job. And the private jets would likely be quite far down that list. (Except Teterboro, that's its own animal and no idea how to manage that through a shutdown except grounding flights).
Agree 1,000% with this.