Battery light

The generator was a good suspect if no testing was done. Problem is about half of the time a charging system problem is caused by something that is preventing the generator from doing its job. Your mechanic needs to look at wiring problems and perform some voltage tests. You should have returned to the shop immediately. The mechanic should have performed a system test or at least seen the "Battery" light when he was done. Either one would have shown him his diagnosis was wrong.

Replacing the battery along with replacing the generator is a good idea unless it is less than about two years old. Some Generators, GM's in particular, develop huge voltage spikes that can damage the internal voltage regulator and diodes, and interfere with computer sensor signals. The battery is the key component that absorbs and dampens those spikes, but they lose their ability to do that as they age and the lead slowly flakes off the plates.