Hmmm.. for the LED winker to go off tends to imply the power to the display board has been interupted - even after player crashes the winker runs. If a button push (getting worse with time) causes a power off, then it sounds like the force of the push on the button is causing some sort of short/OC condition. I'm inclined to say short: if he pulls the player to get it going again it could be a re-settable poly fuse being tripped.

If you pull the player out when it does this, sniff the grill aperture near the power connecter rear right of the case. Is there a faint "electrical" smell?

Can you do this with the mains adaptor plugged in, or is it only in the car sled?

If it's only in the car sled then I am inclined to say there might be something up with your sled wiring that is ticked by you doing this. Not honestly sure.