Okay, that certainly sounds like damage then. Most capacitors are pretty easy to solder and extremely cheap, you can buy them in small quantities. I would try a repair if you know what value capacitor

No, you can’t short the pads. If there’s no capacitor there then there likely never was unless you can see obvious damage. Not all components are placed when manufacturing. A picture would help but it could be a decoupling capacitor which has been deemed unnecessary or not required for the model. So it’s unlikely to be the issue. It could still be other capacitors that are getting old but it would be the electrolytic ones.

I believe I see a relay, capacitors, diodes and an inductor. Nothing beepy