Using an SPST knife switch to power on/off a PC
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.