I assume you've checked all the wiring and connectors?
Is that a model with a rubbery power button if so a likely culprit is that the conducive material from the back of the button has accumulated on the contact causing it to randomly fire. This is the same as happens on tv remotes etc. Remove the buttons and clean both buttons and contact with isopropyl alcohol.
If not a rubbery button model I'd look at wiring it up with power only and see if it still turns on and off. If not you might have an intermittant short somewhere in your speaker wiring such as in the area where the cables flex in the doors, or maybe a speaker coil is shorting out. Or maybe theres a problem with the permanent live.
Other than that I'd be checking for bad solder joints particularly around the connectors where they get wrenched. Check for moisture damage corrosion etc. Disconnect all the internal connectors, ribbon cables etc and clean with contact cleaner, clean power switch with contact cleaner, desolder and swap power switch with another switch.
Usually its a simple thing but these modern radios are microprocessor controlled so it may just be scrap.
