If you knew you’d be able to check your too big to carryon bag at the gate for free you would never pay the cost of checking it before security (unless it is too large to fit through the X-ray scanner at TSA).
You can solve that by not offering it as a default option, but rather only when the need arises.
Here's a case:
Say you have a flight booked at 125/120 seats, there's 80 spots overhead:
- expect 5 not to come (120 left)
- 25 have booked checked luggage, offer them to hand in their carry on (95 left)
- for the remaining 15 until enough people take the offer: offer a higher quota when they check in their hand luggage, e.g. 12kg instead of 8kgs as long as the luggage is the same size
- if anything remains only then use the old system
pretty sure this would be popular, it's less unfair/annoying and takes care of issues in advance at a fairly low cost (of the 15 getting extra weight, not all of these would actually pay for more luggage or even bring more, by not taking more or putting heavy items in jackets etc.)