I spent some time working with that approach of using LLMs to generate synthetic labeled data for use in training more specialized models. It mostly didn't work.

The problem was that getting the LLM to generate training data that sufficiently resembled real-world data was labor intensive and expensive. More labor intensive and expensive, it turns out, than just using real data.

What worked better was using the LLM to label the training data. But even there we had to be careful about introducing weird biases.