Why would this microphone only work while touching the volume control?

If it is a grounding issue, touching it could change the amount of hum, buzz or interference.

However you say "It only works if you keep your hand on the volume knob" which I interpret as "there is sound only when the hand is on the volume knob, otherwise there is no sound."

If this is the case, then I'd suspect a bad contact perhaps at the pot wiper, or a cold/broken solder joint that makes contact only when the volume pot is messed with.

You can check this by touching the volume pot with a non-conductive object like a plastic rod. You can use the shell of a ball point pen, or a toothpick for example. If wiggling the knob around with a non-conductive object results in the sound turning abruptly on or off, then you can rule out any influence from your conductive finger, it'll probably be a bad contact.

If the failing contact is at the pot wiper, it will depend on the pot's position. So if the problem does not depend on the pot's setting, it is more likely this is a bad solder joint on the board.

Repair questions are off-topic on this site so let's make it an education question.

Figure 1. Potentiometer terminals 1, 2 and 3.

^{simulate this circuit – Schematic created using CircuitLab}

Figure 2. A schematic showing the potentiometer arrangement. As the wiper (2) is adjusted from (1) to (3) the volume goes from zero to maximum.

If pressing on the potentiometer causes the system to work then it's possible that the wiper has lost contact with the resistance track. You can test this safely by turning the knob to mid-position and connecting (2) to (3) with the tip of a screwdriver. If that works the potentiometer is faulty.

If you're feeling brave you can desolder the potentiometer, bend the tabs folded underneath and disassemble. You might then be able to bend the wiper a touch so that it springs onto the resistance track more firmly. A spray with some contact cleaner would be good if you have access to it.

The First Photo "looks" like it's broken...

But assuming it's not, this behaviour looks like the potentiometer (variable resistor) is broken. The pressure of the finger pushes the wiper contact on to the track completing the circuit.

It's also possible the solder joints are "dry" and need to be reflowed

Otherwise replace the potentiometer or the whole device would be the best advice.