issues with intermittent powering up and/or spontaneous rebooting are usually to do with electrolytic capacitors (the paste in them drys out) or voltage regulators.
I've had similar issues with other equipment and I just replaced any electrolytic caps and voltage regulators I could see on the board - starting with the PSU. It's hard to test caps in circuit, so it's easier to simply replace them (chances are they will go at some point anyway).
May not be the issue here - but the initial symptoms sound like it