Well, that was an irksome couple of hours! But with a new set of switches, my oven works like new.
Buying a 24” screwdriver from Harbor Freight for nine bucks was a very worthy investment for reaching those infamous black screws. I have no idea how they actually built that - I guess the control panel and case are assembled first, then the actual oven box and heating elements are slid in after?
Now I have 118 switches left over. At least 6x6mm tactile switches are plentiful - doing a board swap is currently impossible as the boards are not available from Breville or the usual aftermarket repair outfits.