Will a processor socket with bent / broken off pins still work if the processor still sits in it ok? Purely hypothetical.
The pins are there for a reason so if any are missing or making poor contact then it's likely to cause some form of problem - permanent or intermittent.
Just for elaboration: you may get lucky if the broken pin/socket is a ground or power one as there's more than one of those, but the power and ground contacts are spread out to share the power load and to cut down on electrical noise so the CPU may still not work properly. If it's a broken data, address, timing or control contact then you're probably out of luck.
If any pins are broken off then the processor won't work - there'll be a missing connection. As with an electrical or data connection if it's not there then the processor can't work.
If any pins are bent, then as long as the processor still fits in the socket (i.e. the pins aren't bent too much) it should still work as there will be all the connections present.