To the title: Yes.
My question is, could I have killed the capacitor by swapping the lamps frequently and starting them up?
No. It was just the capacitor fault alone.
It was likely an internal connection fault - lead wire going to the capacitor spool. The lamp was flickering - that mean the capacitor internal connection was "hanging on a thread" and that "thread" just broke and started arcing (so the flicker), eroding the material further away from the broken spot. So then when shutting down and reconnecting the power, the connection was lost for good...