There's not enough overhead bin space. The four options are:
(1) increase bin space (hard)
(2) decrease or eliminate checked bag fee, at least for legal carry-ons (expensive, but this is the option I'd choose)
(3) first come first served (results in the mad boarding dash we see now, with line jockeying and lining up before boarding even starts)
(4) charge for access to the overheads for some class of passengers. UA could have chosen "all non-elites", instead it created a new class of customer that doesn't have access to the overheads, and charges less for that ticket.
The no carry on restriction doesn't apply to elites, so there's no way it can piss elites off. If elites are pissed off at BE, it's for other reasons (no PQM, no seat assignment, etc.).
I'm 100% sure that the same people complaining about BE not allowing carry-ons are the same people who are complaining that they had to gate-check a bag when they came off a tight connection.