It seems that, to satisfy all requirements of safety (weight), fairness (enforce the rules for everyone) and aeroplane type, we need to have three checks at the gate on boarding.

First, we need to establish whether any bag is over size for UAL's policy. If it is, then it needs to be checked into the main hold and the passenger charged the normal bag check fee PLUS a convenience fee for not checking it in properly (say, a $100 penalty).

Then we need to do a size check related to the specific plane type, with oversize pieces then being gate-checked free of charge.

Then we need to weigh the remaining pieces to ensure that the overhead bins are not overloaded (although I don't see how this can prevent one bin being fully loaded with the heaviest items and another simply being full of coats).

Once all this is done, we will have a plane that is not overloaded with carry-ons, the remaining boarding process will not be held up by imbeciles with three or four huge bags and elite passengers won't have to be lurking around the gate area 30 mins before departure, desperate to ensure they get on first.

All these outcomes are great but the procedural cost to get there (twice the number of GAs per flight???) will outweigh the $100 fines in terms of costs and would otherwise be quite impractical. Shame.