The GEN light depends on the original regulator to give proper indication. The regulator can be reinstalled for partial operation to get light working as before. Another way to monitor alternator output would be to add a voltmeter as used in modern cars and disconnect the light.
Since the alternator is new and everything is self-contained, it should be charging if it's spinning. Check the voltage on the battery with car idling. Reading should be 14 V or so.
The carburetor starter switch ties into the circuit in the same area, but does not affect the GEN light. One of the safety backup features for the starter depends on the original generator for operation. That feature is now eliminated with use of the alternator. As a substitute, a starter emergency kill switch can be added in case starter doesn't drop out after engine fires.
What has been, can be again. (Bob Wills, 1942)