Bob
You need to find out what's gone pop ..
Get someone to meter the motor out, no point lashing out loads of shekels if you have no need to.
If a motor won't start, but runs after a quick flick, it may well be that the start winding is shot. But other, cheaper, things can be wrong.
ie, start switch crudded up and not making contact. Cap gone open cct. Sometimes the start switch assy. gets corroded.
Have a damn good shufti at any rivetted connections on it.
As you have the motor running, can you hear a click as it nears stopping. This is the Start Switch dropping back to the closed position.
I don't know that I could verify any wiring from here ...

I have enough grief with my own, and that's under my nose ..

Dave BC