Electronic – How to know what power supply to use with a circuit

If the switch is connected to a pin of a microcontroller, then the lead will have very little current, and will not be able to drive a relay or otherwise be current-sensed. However you should be able to sense whether the switch is pressed or not by looking at the voltage of the switch terminals.

With the switch open, measure the voltage on the two pins. One might be ground (0), and the other a voltage like 3.3v or 5v, usually referred to as Vcc Then close the switch. Depending on whether there is a pull-up or pull-down resistor, one of the leads will change to match the other (both will now be ground, or both will be Vcc). You can then use that lead as the indication of your switch closure.

As you proposed, you can use a solid-state relay across the contacts to simulate the switch closing.

Before you do a lot of debugging test all the other soldering places as well. I think maybe somewhere else it could have connection problems.

Maybe put an additional cap of about 100nF after your buck. It's better to have also one with a small esr. Touching the cap shouldn't reduce the voltage to 2.2V, I guess this could be a problem.

When you haven't found anything in the hw then there's maybe a hold state for power on. Have you unplugged your debugger and programmed the chip? Maybe he is keeping the chip in reset state until your voltage drop.