Maybe one of the guys who has actually made this repair will step up with specifics but in the absence of that, here's what I'm thinking.
If I understand what others have said, the simplest solution is to solder wires to the two power switch terminals on the upper left. Then run those two wires out of the box to another "beefier" switch and you're done.
OTOH, I can see how this is one cool layout accessory and if it were me, I'd really want to replace that switch so everything fits in the box and works as intended. So, if anyone else wants to carry on the discussion, it appears the board is held "permanently" in 5 points.
1) power switch with two terminals in the upper left of photo
2) up/mid/down switch
3/4/5) up, moving, and down LEDs
I believe the up/mid/down switch can be released from the box by un-loosening the nut. I believe the on/off switch has to be released from the box by removing the solder from the two switch contacts of the board. If you were extra-ordinarily lucky, the board would then come free from the box. and you could access the switch. But the wild-card are those 3 LEDs. It appears they used so-called "LED clips" which are shown in the inset. Once the LED is snapped into the bezel and then the threaded black retaining ring is pressed in place, it is impossible to pull the LED out from the back without removing that retaining ring (which is really hard to do considering the angle of entry you'd have). So if you are really ambitious, I'd say you'd have to remove the solder from the the 3 LEDs (3 x 2 = 6 connections) using copper-desoldering braid or a de-solder vacuum pump.
Yes, all this sounds very time-consuming but I really think the board can be removed though as the #1 ranked contributor to this forum will remind me, nothing is so easy as the job you imagine someone else doing. Seriously though, I believe the board could be removed in, say, 15 minutes would require soldering finesse.
Actually what I'd really like to see is the failed switch - maybe take it apart and look at the contacts or whatever.