they should have installed a motor starter not a lighting contractor .
The "just a loud hum" with no rotation at all strongly suggests something much worse than voltage imbalance, like single phasing. If the motor itself is in fact good, that is.
And that old lighting contactor has probably been there as long as the original motor. It won't cut out like a motor starter and will let the motor draw current until the breaker trips.
If it were me I would want to make sure I had the correct power before hooking up a new motor and possibly burning it up to. He knows he is not getting correct voltage but jumps to the conclusion that the load is not balanced causing the problem.
He never says if he reads this voltage imbalance back at the panel.
People may not think about but the reason they make rebuild kits for motor staters is they do wear out.
I'm not saying the contactor is bad but I would check it out before going any farther. I have replaced a lot of contacts in motor starters and the probably got more use than those controlling an exhaust fan but I found bad contacts at an air compressor a couple of years ago.