I have a strong suspicion that yes, but as I am not a specialist in o-minimal structures, I thought that I might have overlooked some corner case.

The precise statement is as follows: let $X \subset \mathbb{R}^n$ be a set definable in an o-minimal structure. Then there exists a point $P \in X$ and an open neighbourhood $O \subset X$ of $P$ such that $O$ is a closed analytic subset of a real analytic manifold embedded into some open $U \subset \mathbb{R}^n$

cannotget too wild, but this is a top-down argument quite orthogonal to how the functions were generated. Basically, I can imagine this to work only in presence of powerful quantifier elimination, but then the result would not really say anything interesting. $\endgroup$