Shuting down\off after several minutes of operation.

You can also check that the back of the motor is clean of debris. i have the same issue but i recently cleaned the back of some cat hair, not too much and it seemed to be ok now. i never had this problem until about 6 yrs after i bought the fan so i know it isn't a heat issue or sensor control issue or it would have done that from the start

This thermal switch may be of the type where it is saving power by turning off motor when air is sufficiently cooled. You obviously want the air cooler, so you can place a wire across the two terminals of that switch to defeat the power save. If the switch is to prevent motor over-heating, however, then you cannot jumper that switch. Do this test: open back to locate switch. if switch is located directly on motor housing, then that switch is cannot be jumpered out. if switch is located in air stream, then it may be possible to jumper it out. Try the jumper and observe for many operations of fan useage to ensure jumper is safe. Ensure no bare wires or electric terminals exposed when done.