There is only one real solution to the “Rome problem” of too many sites and the necessity of tourist triage.
Extend your stay.
And go back.
The top of my must-see list in Rome would be the necropolis level under St. Peter’s. It is an inherently limited access space because it is small and the surviving frescos are vulnerable to moisture, so the number of daily visitors is controlled. Despite that, it does not require any special hoops to visit; just make reservations early to be sure of getting in. And visitors should read up a bit on the archaeology related to St. Peter’s burial site before going. That’s a good thing to do anywhere, but it’s close to essential in the necropolis. The guide will explain it, but the details are sufficiently intricate that a little prep goes a long way.
Thanks, good idea!