Lovely dryer. I have never seen one of this version, only ever seen timed dry not auto.
1. I see a burned wire and a scorch mark in photo 2 of reply #3. Brown wire indicates active wire. So you may have had a short to earth where that burn mark is. This may have damaged the electronics. I don't see evidence of wire rubbing through, may have been a mouse bite a wire??
2. I think you have to separate the front panel from the rest of the dryer - those small black hex screw heads in photo 1 of reply 3. That should give you access to the screw or screws holding that circuit board.
3. Have you ever reassembled one of these Simpson dryers? They are a pain in the arse. You have to thread the belt through the tensioner and around the motor pulley. If the belt is even slightly out of alignment, the moment you rotate the drum, the belt will slip sideways and the tensioner will ping off. The drum will have a black mark around it where the belt has run, DON'T clean it off. Use that black mark to show you exactly where the belt lines up. If the belt is a couple of cm out of whack, the tensioner will come off when the drum moves. I find Hoover dryers of the same vintage much easier to work on.
4. The sheetmetal edges where you will have to take off the front panel may be SHARP. Be careful, wear leather gloves. I severed a tendon in my hand on a similar edge many years ago. (Actually on a UK Hotpoint washer.)
Before going in too far, are you SURE the element is staying on? from vague memory these dryers take quite a while to cool down. If you have a meter to monitor power consumption, watch how many watts the dryer uses during heating and during cool down phase. Should be over 2000 watts heating, drop to maybe 200 to 300 for motor only.
If the heating is definitely stuck on, are you able to do circuit board work? I may be able to help if you like. I'm not a professional, just a competent amateur. I doubt the board would still be available. I'd suspect a fried transistor or a stuck relay.