John,
The fact that it can run says there is nothing basically wrong. These are not brush type motors however there is a centrifugal starting switch that can get cruddy over time and all it takes is a small flat file or fine sandpaper and some time to clean the contacts. This switch is usually behind the back cover where the power line goes in so if you can remove that cover you will see it. From your description it sure sounds like the switch isn't closing when the motor is stopped and when power is applied you get a little vibration, humming, that causes it to close and start the motor.
Hope this helps and saves you $$$
Al