Electronic – Using SPST Switch as Button using Capacitor, which type

Yes! It's possible for mechanical encoders with detents. It's going to limit you to a fairly low number of steps per revolution because the mechanical tolerances get troublesome.

Consider this CUI part (photo from Digikey):

The detents are such that the encoder outputs are guaranteed to be 'open' at the detent positions, so your pullups will draw no power (the vertical dashed lines indicate possible detent positions).

I was really hoping to only connect the circuit to the PC's power switch wires, and not require a separate power supply line.

Is something like that even possible?

Converting a toggle or latching switch, into a momentary output, typically requires some active parts, like logic ICs, a 555/556 timer, a microcontroller or dedicated IC.

Your capacitor + relay circuit is one of the simpler methods, but the pulse will be equal length on both connect/disconnect of the knife switch.

A microcontroller with a simple interrupt code would be easy to setup for your requirements, and only need the microcontroller, a capacitor, and power.

Honestly though, you are already running two wires, one of which is either ground or VCC, a third wire would be trivial.