Theory Question: Speakers don't work unless I poke soldered points with screwdriver
Most volume controls have the terminals rivetted to the resistive track. Turning the volume control up and down a few hundred times (especially if it has the in-off switch in it) or pushing and pulling it loosens the rivets and the resistive track disconnects until something (your screwdriver) moves it a little.
My Sony Walkmans and clock radios had that problem.