>>> what do they do with passengers who are within the limits for carry-on baggage but can't stow their bags on the plane because the bins are full ...?

A: The stuff will have to be put under the seat in front of you. If such space does not exist for example you have a bulkhead seat, then you will have to trade with someone.

A: (really) If your stuff is within the limits it will still be gate checked if absolutely positively necessary since you have already cleared the first screener at the top of the jetway.

You may have read other threads on Flyertalk describing FA's who seemed to work wonders with fitting everything into the cabin. Meanwyile culling the pax who had too many or too large items at the top of the jetway speeds up the FA's job and helps an on time departure.