Yes due to overheating.
I don't think this is correct, this isn't why hash boards with no heatsink don't work.
But my question is: if a heatsink falls off, is it normal for the hashboard to stop working?
yes, once you fix those heatsinks, assuming all the 63 chips are fine then the miner will work just fine, there are many videos on youtube that show you how to glue a heatsink on those chips, it isn't easy, but if you have done something similar before - it shouldn't be difficult.