Like Matchless said, probably just replace the capacitor. You cant check those big ones with a normal meter you have to have a special one. Probably any A/C repair shops in the area could check it. But its likely just 6 to 12 bucks to replace anyways and more often than not they are the culprit when it just wont start up. Sometimes they look messed up, but other times they look perfectly normal but are bad.
Loose connections can also lead to heat, which might be why the connectors got hot enough to melt. Otherwise a large amperage draw was the culprit, but I imagine you would have noticed the lathe laboring.
Those contactors dont look that trashed to me. Just clean gently and they should be ok.
But like you saw that switch has got to be replaced too. Its probably not easy to get replacement parts for a Chinese lathe either, just gotta get what you can. If you need any translation I actually speak Chinese. ha ha.
Bummer, its always one thing after the other isnt it. And way to often around tax return time.