Electronic – Risk of long-term shorting XLR cable

Using a pot as a voltage divider as you do is not a good volume/gain control. The problem is that it makes your output impedance dependent on the gain. This has the effect of making your frequency response vary with gain as that output impedance interacts with the capacitance/impedance of your cable and whatever you are feeding this output into. You want the output impedance to be as low as possible. Do this by deleting R5 and R6.

To do the gain control without impacting the output impedance use a pot to replace either R3 or R4 (or both).

The nice thing about this type of gain control is that it actually controls the gain. With your configuration if the opamp was clipping and you turned the gain down you would still be clipping the opamp. But with the modified gain control, if you turn down the gain the opamp will stop clipping.

Do not remove C2, since it blocks the DC signals to your sound card. You might even consider increasing the size of C2 to as high as 68 uF. The larger C2 is, the better your low frequency response will be. 10 uF is the lowest I would go, but 22, 33, 47, or 68 uF are better choices.

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.