Pump hum and delayed start

[Pappy]

Well, since after it hums then stops, then starts up about 30 seconds later just fine all by itself, doesn't that mean that there is nothing impeding the impeller?

Not necessarily. Many times things move more freely when warm than when cool. This happens because different materials expand at different rates. And I also recommend changing everything, if you go that route at all. If you have access to an ammeter, check the amp draw, both when the motor is humming and when running and compare that to published specs.