Reviving this thread again. I had a couple of ideas but none of them turned fruitful. I returned the Amazon motors and bought a new pump from InyoPools (same 3/4HP motor with 1.27 SF, for a total HP of 0.97). It did the same thing - shut off on its own after 10-15 minutes when the motor gets hot. I thought maybe it could be an issue with the wiring from the timer to the motor as it was super old and the carflex was cracked in multiple places. That also didn't solve anything outside of the comfort of knowing the wiring is new and prettier now. I'm pretty sure I've ruled out pump motor issues at this point since this would be the 4th motor I've put on it (2 from Amazon, returned those two, bought one from Inyo it did the same thing so exchanged it (again through Inyo) for a new one that also did the same thing. I suppose it's not completely out of the realm of possibility that both motors from Inyo are bad though...
I bought an amp probe and found that the motor is drawing 15-16 amps under load, whereas the label has the amperage at 13.4 (I also just realized the label says "max load 6.7/13.4", which I believe tells me that the issue is the amperage draw is too high, causing the motor to overheat and go into thermal overload protection (tell me if I'm wrong - just thinking out loud). I also decoupled the seal plate and ran the motor just long enough to get an amperage reading and it runs at 7-8 amps under no load. Once recoupled, as the pump primes and pressure builds, it steadily climbs back to 15-16 amps.
In the same vein of making old things new and solving simple things, I also went ahead and replaced the 20 amp breaker on the outside panel and the 30 amp breaker on the main panel in garage. Again, neither of which solved the issue but at least they're new and have been ruled out.
Thinking there may be a blockage in a line and trying to cover all of my bases, I got one of those clog removal bladders that go on the end of the garden hose and back flushed the skimmer, bottom drain, and vacuum lines, each individually - all of which "came back" clean - as in, no junk came out and I could feel the water coming back up the line (with the exception of the bottom drain because the pool is currently cloudy and cold and I can't see it from the top. The two jets also blow clean except the furthest jet still has small air bubbles when the motor is running (like fizz from a coke can, not large gulps of air).
I replaced seals on the pump basket lid, filter lid (Sta-Rite PTM70 cartridge filter), and the air bleed screw on top of the filter. Pinch-A-Penny recommended not messing with the bottom seal but I'm tempted to replace it anyways.
In replacing the seals, I did find a pinhole leak on the top of what they are calling the tank body (the other piece that the seal plate mates up with). I haven't plugged that and I'm thinking some JB Weld will do the trick there. Other than that, I don't see any drips or moisture anywhere else and don't hear any hissing, but it's also right at the pump so that could be masking it.
Is there anything else I should be checking?