I hate to state the obvious, and don't mean to offend anyone, but you say when the valve is turned on it continues even though you stop pressing the lever. Did you change the filter recently? If so you will need to flush couple of of gallons through the system to remove air, I've been out on a lot of service calls where that's the only problem.
There is a button somewhere one the control panel that you press for three seconds to lock and unlock the control, it is a child lock. The is also most likely a small picture of a padlock on the button.
I liked the original Fisher Paykel dishdrawers. Easy to diagnose and repair, and generated LOT$ of COD service calls. The later "enhancements"......Feh. I'm still having nightmares about the F9.
Ya, that F9 is a pain, it can be pretty much anything. I've had a few of them and each time it's a different component and the only way to find it, at least the easiest is to pull out the full service manual and ohm out every single component from the board.
Try turning the impeller in the wash motor while it's pulled, if at first there is quite a bit if resistance that us most likely the problem. If that works and it reoccurs frequently then you need to replace the motor, very $$$.
If you ran the on board diagnostics it will not give you a definitive answer if the thermistors are good or bad. You would need to actually take some resistance measurements at the control to verify that the thermistors are good.
If it tripped. It tripped for a reason, that's why in certain situations they use non-resettable bi-metals. They don't want someone to just pushing the reset button without addressing the problem.
Also, the resettable bi-metals wear after getting reset from thermal fatigue and arcing of the contacts and may not respond properly in the future.
And, also, the are bound by UL to use certain types of safety thermostats in certain places.
Also, again, no manufacturer wants to put money in the pockets of the repair guy. If anything, an expensive brand like FnP wants you to have a good experience with their product that way when you need a new one, for whatever reason, you purchase another one of theirs.
I'm not critiquing your work, plenty of ingenuity.