One of your pictures shows an internal capacitor - or was that the picture from the internet that did not match your motor? There is a capacitor on the side, I take it; does it have the value indicated (μF)?
Describe the symptoms again: does it never really start (just crawl along), or does it start but not come up to full speed? Lots of induction motors use a single start capacitor which comes out of the circuit when the centrifugal switch releases. If the cap is completely bad, it likely wouldn't start; it would just hum. Same is true if the centrifugal switch did not close. However, if the centrifugal switch doesn't open when the motor starts to spin, I'm not sure what the symptoms would be (definitely overheating as mentioned above) but in any case it wouldn't run right. I'd take a close look at the centrifugal switch; I'd also look at finding a similar start cap and trying it in the circuit - if you know anyone who works in HVAC, you may be able to browse the scrap bin and have your pick of both run and start caps.
