I can spin up a classification solution with an LLM in ca. 5 minutes that is likely to do an okay job. To build a custom classifier, I need lots of training data first.

That is the reason.

Once you then have a system that is used enough, it starts making sense to look to replace that with a leaner classifier, but LLMs have a place in vast number of places where we'd previously never use a classifier because it was too much effort.