Yes what you're describing is a classic way of doing query understanding.

I've found, though, getting it in the language of the vocabulary has generally improved performance.

Further, when searching for "blue shoes" you want to separate the color from the item type. So its useful to have a dumb LLM do this for you. And with the LLM in the loop, its further useful to get it into the language of the taxonomy to improve embedding retrieval accuracy.

There are of course many ways to skin the cat here :)