My local bus system has ramp entries, and the driver can require non-mobility-impaired riders to give up seats at the front. (And those seats can also be folded up to make room for wheelchairs.)
Would something like that work for the New York subways? Or would it require too much oversight to make people follow the rules?
Simply wouldn't work - each train has a driver and a conductor, the driver does not interact with passengers at all (and for safety reasons, probably shouldn't), while the conductor can only oversee his/her immediate surroundings.
The issue goes beyond simply wheelchair-bound passengers, but also the fact that the elderly also ride the trains. There are a large number of people who aren't fully disabled but for whom having a seat is important, so it's not enough to simply reserve a few places for a few individuals.
The 4/5/6 trains are a clusterfuck because they're vastly oversubscribed. The serve the entirety of the Bronx and the hyper-dense Upper East Side, not to mention the entire east side of Manhattan. The solution to this isn't "leave a large demographic of riders in the cold to gain marginal capacity", the solution is "build the f'ing 2nd Ave subway".
There are a large category of improvements to the subway that, while valuable, offer only marginal improvements to capacity. Open-gangway trains can increase per-train passenger capacity by ~5%. CBTC digital train signaling upgrades will get you ~10-20% more capacity by packing trains closer to each other safely. Even stacking a bunch of these improvements together will not resolve the massive capacity deficit the MTA suffers from system-wide. Only new track, new trains, and new stations will.
IMO standing-room-only cars fit in this category. It would give us a bit more capacity, but nowhere near enough to make a real dent on the capacity problem - and this particular "improvement" has a high cost in excluded passengers.
Even if we made all the cars standing room only, you'd still be packed like sardines into the 4/5/6 in the morning, and you'd still have to watch helplessly as multiple trains pass by packed to the brim.