No volume on the same side as the buttons and controls.

Also it sometimes shutoffs while using Bluetooth or the USB wireless adapter.

Open it, the really tiny wires leading to speaker often just fall off due to rush factory soldering

That's planar mag drive; if the wire is indeed torn, soldering it would not be an easy feat; but chances are, membranes got torn or creased, this particular model has quite crappy material according to a quick search. You might want to unsolder the wires from control circuit and attach some other dynamic to their terminals to check if there is signal; also carefully measure resistance between the same wires (should be like very low) to check this hypothesis.

With broken sound making component, the most sane approach is to buy a new acoustic driver. Rolling out DIY means pretty much designing it (or reverse engineering the old one with unknown materials), matching sound geometry and impedance and all, which is kinda hard, time*money expensive, and fun.

Volume potentiometer has one channel broken?

Have to add, after reading the name of the device, I think I know what joke is coming, just a matter of time...

Where do you see the wire is torn? All the wires look intact to me.

I do not see any breakages, it might be torn right at membrane that is not visible on these pictures. Quite common failure point. Getting there is quite hard, just testing the line for continuity might be simpler and would rule this out or not.

Do I need to remove some of the protective plastic film in order to test the line like you said? How could I re-protect the metal links afterwards?

I think the easiest way would be just to unsolder the wires from the PCB on the first figure. That's what I'd do first and safest.