Most likely this is an input filter capacitor for the 12V rail of the PCIe slot. It's used to reduce the voltage ripple of the power routed through the PCIe slot and is probably not super important for the GPU core itself since you will have 6/8 pin connectors with their own filter capacitors directly on GPU PCB that should handle most of the power draw. A possible outcome in this circumstance would be reduced OC potential for your core/mem clocks due to the increased input ripple if voltage was your limiting factor.
Depending on the circuitry and layout for the GPU, the 12V from the PCIe slot may be used to generate additional voltages that control different features of the GPU and circuit behavior could be changed by larger then usual voltage spikes/dips caused by the lack of the capacitor; albeit unlikely due to bulk capacitors primarily suppressing voltage ripple from large changes in current that just don't happen with most of the supporting voltage circuits.
TBH, I would be more concerned about traces getting damaged in the general area where the capacitor got sheared off than the actual capacitor itself. Most of the symptoms you said sound like software issues or a bad OS install, but there could always be some kind of underlying damage to data traces that report the status of the device installed and could cause all kinds of funkiness. I personally would try re-installing your OS before concluding anything about the monitor issues with Nvidia drivers; especially if the same issue happens with the GPU in an undamaged PCIe slot.
Regardless of what happens, I would recommend you to install a new 16V 270uf polymer cap of similar dimensions and low ESR/ESL in that location before you use that slot long-term to limit potential damage to your GPU. Better safe then sorry!