Been watching this, I wanted to jump in but been having some health issues. Mostly not being able to breathe. But I think I've got a handle on it right now.
But, if you don't get a reply to a question, just be patient.
No way to tell if the start windings are bad from the pictures. If they are not shorted to ground, then that is about the only thing you can test with your equipment.
But, I have seen some a lot more burnt than that that ran just fine.
First thing, replace the starting cap. If its leaking, its bad. With it bad, the motor will either not start at all, or never get up to full starting speed.
That weight and springs that should be on the back end of the motor shaft works the switch. Lift the weights against the springs and make sure the washer looking thing moves smoothly.
Look at the board at the end of the motor with the two arms with nylon pads on them and make sure the contact is making contact.
When the shaft gets to a certain speed, the weight swings out, and the washer releases the two arms and opens the contacts. This takes the start cap out of the circuit and lets the motor run on the run cap.
The run cap usually will not show any obvious signs of failure. If one if the contacts is leaning to one side, then its bad because an internal fuse has blown.
But, not all of them will do that.
You can test the cap with your meter to a certain extent. Disconnect it, carefully short across the contacts. Do not do this if the motor has just finished running.
Put the meter across the contacts on ohms. You should see a very low reading that will quickly ramp up to open circuit as the cap charges.
Since that is a small value cap, it will charge up quick. A digital meter might not catch it.
I doubt that the windings got that hot that quick. With the start cap leaking, it has probably been having problems starting for a while before you put it up.
As the cap weakens, it takes longer and longer to get up to speed, and the windings get hotter and hotter.
I would check the switch, make sure its working, replace the start cap, and see what happens. If the run cap is bad, it will usually run, but not cleanly and will make noises.
Given the price of a new motor, a new start cap ranging from $5 to $11 depending on where you buy it, is a good gamble.