If it has a Low/High charge rate control resistor on the light switch, it connects to the gennys FLD post and then case frame ground (if in LOW charge). Its purpose is for HIGH charge if the gennys FLD post is dead grounded (i.e. resistor isnt in circuit) but LOW charge if the field control resistor on the switch is between the gennys FLD and case frame ground.
If the switch has no case frame ground path/connection that resistor isnt passing any current so its cool plus theres no charge. If the switch has a good ground and its set on LOW charge then that resistor is passing current and it gets hot. If the switch has a good ground but is in HIGH charge the resistor isnt in the current path and again its cool.
If you go down to the genny and just use a jumper to dead ground the gennys FLD post the genny should high charge. If so then but NOT when the switch is in the circuit, the problems either the switch has no ground or the wires bad from FLD on genny up to switch or the switch is bad or miswired
Run my troubleshooting procedure below will tell how to test the genny and how to tell if non charging is a genny versus a VR or switch problem