Thank you for your question and I understand your concern.
I would suspect a faulty belt/idler spring switch or the drum belt has come off or has broken. If the brown wire between the start switch and the broken belt switch has come loose or if the pink wire comes loose, the motor will not remain running when start is released. If the centrifugal motor switch does not close contact M5 and M6, the motor will not remain running when start is released. Check the belt switch operation and you find it not closing causing the symptom you have. You can test the idler/belt switch for continuity with an Ohm meter.
I added an image below of the wiring diagram and an image showing how to remove the drum.
NOTE: Disconnect the power cord from the power source before servicing.
If you do not feel confident repairing this problem yourself, then you can have it repaired at your home by a Sears technician. Here is a link for the website:
Sears Home Services