Over current trip or overheat trip ?
Can you turn the chuck and gearbox freely, or are they stiff or partially seized ? Might be a bearing binding up somewhere overloading the motor. See if you can run the motor disconnected from the drive-train. And vice versa.
As above plus, some motors also have a bi-metallic switch in them which cuts power when things get really hot. Unlike an overload protector the motor holds the heat for ages.
I had a 3/4hp General Electric motor with one mounted in the end cap along with the centrifugal switch, it cut out after 20 mins running at 60C with a 5C ambient when the motor was sucking 10A+ instead of less than 6A.
If the cold motor starts and spins up normally and the centrifugal can be heard clicking off within a few seconds (you may need to take the belt off to reduce the lathe noise enough to hear it,) then the start circuit would seem OK and it might be winding damage.
Caps tend to show slight bulging or weeping when bad and some motors have a both a start and a run cap to complicate things. The motor plate should have the details on it.
All roads do lead to the motor and these days there are standards for motor mounts, but the shaft size can bring drive pully issues.