I've got to agree with a lot of posts here-
First, if the motor spins easily by hand, the bearings might be noisy, worn, and generally need replacement (or they might be fine), but if they cause no undue resistance to turning at low speed, they are not causing a starting problem. That problem is going to lie in the capacitor, the wiring between the capacitor and the start winding, or the start winding it's self. I would start with those. As a whole, motors are a little bit intimidating if you don't fully understand them (as I certainly don't), but if you break it down to one piece at a time, it's not so bad.
Second, the circuit and the extension cord..... Yeah.... If using that cord is causing problems, that cord is a problem. Even if it "works OK", if it's so undersized that it's causing issues at startup on a working motor, then it's undersized. That means that it's causing the motors to run at a low voltage when they're running. To put it nicely, that's not good for a motor. And while you havn't offered any proof..... My spidey sense is tingling a little about the outlet you're plugging it into. I have visions of a light duty circuit, maybe backstabbed wires, being used for heavy loads. If that's going to be used for this on an ongoing basis, I'd recommend looking into that, maybe measure the voltage drop when starting or running a motor under a load, or at the very least pull the outlet out and make sure it's 12ga wire and not 14, and that it's screwed to the outlet, not backstabbed. The machines that get plugged into it will appreciate that.