I’d put money that the battery is getting warm expanding ever so slightly and forcing the case into the power button.
If this happens:
1-Can you turn it back on immediately or does the button seem different? Check all buttons.
Yes: It might be chip/SMD issues.
No: Battery.
Every time there is an issue, I inspect, fiddle and observe. Since there is a commonality of battery expanding in the unit and mushing the buttons from the inside, I’d assume that yours is at the level where it is beginning. Most of those I mess with have badly expanded batteries. So I haven’t seen it begin.
I’d put money on corrosion. There is likely oxidation that is short circuiting pins/traces. This corrosion could be anywhere and everywhere…. Even apart of the battery. I’d clean up everything with alcohol and q-tips.
I'd put my money on a chip failure. I click to me suggests that it is preparing to work and is interrupted. If all the solder joints are good and no damage is seen then an inline capacitor tester is a good thing to check caps.
We really need a board diagram with voltage test points to properly diagnose these.