OK KM1 is not latching- so either coil is bad or no power getting to coil (bad connection, open overload relay)
Got a meter? Couple tests we should do. First, set the overload relay dial so it points straight up to 11. Power test the machine again.
If that didn't totally fix it:
Disconnect the lathe from power (pull plug)
Check continuity of KM1 coil and overload relay contacts with ohmmeter
Let me know if you are not sure about the testing- KM1 coil terminals are way in back- you may have to trace the wires forward to reach a point
you can access to take a measurement