Making progress, but still not there yet.

I asked you questions about weight and load, stops, and hills. Add to that acceleration from a dead stop and amount of pedaling. I do not ask these questions out of idle curiosity. I ask them because the answers are necessary to proceed in solving your problem.

That motor is a fairly high-speed, direct drive motor. Verify that cruise speed is 15 mph, what is maximum speed?

Knowing what motor you have, plus the melted phase wires, TWICE, the slow speed, point to a problem but the un-ventilated controller adds a possible completely different issue.

Get the bike in a working state. Temporarily disconnect installed throttle and connect old throttle, test. Rare for a fault to "kill" a throttle, but they are often defective. Symptoms seem consistent.

Overheated controller often flaky. Random shutdowns common. Sometimes they are OK when cooled down, sometimes not. Easy fix, no harm, controller can freeze no problem.

A motor that melts phase wires almost always either has an internal short, an incorrect phase or hall sequence or possibly a bad connection on one wire or a broken wire or solder joint, OR, very, very often, is running too slow with too high a load, causing high draw on the battery, which with weak cells can cause huge sag to shutdown either thru controller or battery LVC. BTW, the "BMS" on those batteries is minimal, at best.

Also, once you have melted phase wires on the OUTSIDE, the odds of having melted wires on the INSIDE go up dramatically.

Try driving the motor like a car stuck in high gear. No throttle from stop, pedal only, pedal hard, also on hills and while accelerating. If you are heavy drop 50 lbs. Check the motor cover by hand for heat.

I should clarify that there is no thermal shutdown in the motor unless there is a temp sensor AND the controller is programmed for it. Not likely with that kit. Most controllers do have an internal thermal shutdown. Also, shorted phase wires, and incorrect sequence, can permanently damage controllers.

Describe in some detail the EXACT sequence of events that takes you from a bike that DOES NOT work, to when it does. Wait 20 minutes, charge, jiggle wires, dance the hokey-pokey, turn off at display, or at controller, or disconnect battery mains? These should all be tested, particularly controller and battery off, but even the hokey-pokey, as well.

Never discount evil gremlins or the off-chance that it really IS what it's all about.