Sounds like it may be a timer problem.

I could not find a wiring diagram for this unit so am doing some guessing here.

My reasoning is that the push-to-start relay must have remained closed (activated) for the unit to start.
The motors centrifugal switch must have deactivated so it had the start and run windings in circuit because it started with no problems.
Your thermal fuse must be OK.
All that is really left is the timer.

Also could be a bad connection or wire.
Check that the connectors are properly crimped to the wires especially at the timer.