For a safe on/off control, look in the hot tub section of the store. Hot tubs have a safe switch operated by pushing on a device that I think is pneumatic.
From this and past posts, I take it a family member is having difficulty with the old twister type valve. Before going all the way into applications like adapting pneumatic or electric controlled industrial or commercial solenoid valves, have you checked into what is available on the market? Looks to me like you need to look at the Kohler DTV six port thermostatic valve coupled with the DTV controller. Might be a little pricey, but it is built for purpose.
Sure wish I was nearby, Bert, a fresh set of eyes might catch something simple that you have missed.
Sometimes when the water drips on a single handle control, folks think pushing it harder will make it stop, like with the old faucets. That just breaks something. If this unit has the little rubber caps on springs there are two sizes of springs! If long ones were put in for replacement with new caps, this may cause the friction you mention.
If the alignment of a tab in a slot is off, you'll get friction. If this is a Moen, it should be assembled with lubricant, and there are special tools to replace and install the cartridge. If the works are bent, you'll get friction.
Going to all this bother for a shower control is a real wast of time. I would bite the bullet, tear out the tile, and replace the whole control.
Some of the new ones have scald protection, too, real nice to have if you're in a situation where somebody might run water while she is in the shower.
Best wishes, I know how it is when you are trying to satisfy a customer.