If the switch has a metal housing and is grounded and connected toLike I said, don't rely on your lathe to be a wire. "Happens to be
connected to ground" is not the same as "grounded".
Every electrical part of your lathe - switches, metal electrical
boxes, motors - should have a ground wire of suitable gauge connecting
it to earth ground through the power cord's ground wire (which is
"grounded" via suitable buried copper conductors, eventually).
If other metal parts happen to connect to these grounded electrical
components, that's only "happens to be connected to ground". Rust and
paint are pretty good insulators.
If you decide you want to ground your lathe, you'd connect a suitable
wire to the lathe itself, and connect that to earth ground somehow.
Note that code does NOT allow multiple earth grounds in one building.