Passengers should be weighed. The accuracy of the CG calculation (and thus the safety of the fight) is more important than anyone feeling "fat shamed".
The actual weight of the passengers and the bags carried or checked by the passengers have no bearing on the calculated weight of the aircraft.
For most flights in the US, every passenger weighs 190lbs in the summer and 195lbs in the winter.
Every bag weighs 30lbs (or 60lbs if it’s a “heavy” bag)
An example of airline math I use:
You bring a legal size carry on - 9x14x22 to the airport. You pack it and weigh it at home on your bathroom scale. It weighs 45lbs.
How much does it weigh on the airplane? It depends.
If you bring it to the airport and check it at the ticket counter, when it’s loaded on the airplane it counts as a 30lb bag.
If you bring it to the boarding gate and valet check it at the jet bridge, when it’s loaded on the airplane it counts as a 20lb bag.
If you board the aircraft and stick it in the overhead bin, it weighs nothing - 0 lbs, because it is “included” in the 190/195lb weight of the passenger.
So it doesn’t matter what your bag weighs or how much the passenger weighs.
That’s the way it’s done for all the flights in the US. Out of all the daily departures for the last 20 years, it’s only been an issue a few times. (It is however unfortunate for the few times it has been an issue.)
Bonus animus sit, ab experientia. Quod salvatum fuerit de malis usu venit judicium.