You can do this. It's just sticking a different classifier head on top of the model.
Before foundation models it was a standard Deep RL approach. It probably still is within that space (I haven't kept up on the research).
You don't hear about it here because if you do that then every use case needs a custom classifier head which needs to be trained on data for that use case. It negates the "single model you can use for lots of things" benefit of LLMs.