Just a little update. I got the Turbo Stick yesterday, and after I played with the remaining turbo knob for a second it also broke, lol.
I just took it apart, and I saw why the Run button didn't work. The mainboard was cracked, and it severed the trace for the Run button. How it didn't kill the Select button as well I have no idea. I scratched the trace at the break to expose the copper, put a glob from my circuit writer pen, and voila, Run works again!
I slid the turbo switches to full speed since I don't have any spare parts to fix them, and cleaned up the case, it's in pretty good shape now! I'm really tempted to try to put in a replacement stick and buttons on it. There's not a lot of room in there, but I think it could be done with stick out of a Neo Geo or Saturn ASCII stick (both are small and microswitched) and some Seimitsu 24 mm buttons.