Noticing a buzzing sound on one start attempt, I knew for sure it had to be a hardware issue. I opened the laptop as shown in this video. Do note that the rubber bands are glued, if you place them sticky side up you can just re-attach them later.
After removing the bottom shielding, I noticed a stray plastic piece, it came from the LAN-connector. It has a spring to keep it close, when not in use. The spring presses against a metal plate, held in place by two plastic bolts. One of the bolt's head had come off, hence the loose plastic. Meanwhile the metal plate was being pushed against the mainboard by the spring and would short it on movement.
Gluing the metal to the remainder of the bolt is enough to keep it in place. So do that and wait for the glue to settle. (While your laptop is open you might want to clean the fan, just be careful not to spin it and cause a short.)
Now clip in the bottom shielding back in place. Test if it boots and can be moved without crashing. If it's ok screw the shielding in. Be careful with screws, not to overtighten then. Don't forget to attach the rubber bands and you should be good to go.