If you want to get some of the benefits of the recirculation system, put the pump on a switch (or even better, a timer switch) near the point of use. Flip it on a minute before you use the water, then turn it off and enjoy your hot water. You're not wasting water or energy then.

Technically it might also show grounded if it's touching a metal box that also is metal all the way back to the panel, or if it's tied into a neutral prior to getting back to the panel.

I would personally stick with the plan of making the first outlet in the circuit a GFCI. You can still tie into the wire that may be ground, and if it's wrong, the GFCI will still attempt to protect you.

I have something like this - my reverse osmosis filter system is fed by the laundry cold water supply, because I wanted the filters somewhere easy to reach, so they're in the basement by the laundry sink, rather than under the kitchen sink. I just run a little 1/4" line up to get to the r/O spigot and I'm good.

The other problem is going to be cleaning up the old threads, since you probably don't know the thread size so you can't pick the right tap. You might be better off picking a slightly larger drill bit and then re-tapping the hole slightly larger. It looks like the valve is behind a sheet of plastic, so my worry would be that replacing the whole valve is going to become a much larger shower/tub project.