The Robertson–Seymour theorem says that any minor-closed family $\mathcal G$ of graphs can be characterized by finitely many forbidden minors.
Is there an algorithm that for an input $\mathcal G$ outputs the forbidden minors or is this undecidable?
Obviously, the answer might depend on how $\mathcal G$ is described in the input. For example, if $\mathcal G$ is given by an $M_\mathcal G$ that can decide membership, we cannot even decide whether $M_\mathcal G$ ever rejects anything. If $\mathcal G$ is given by finitely many forbidden minors - well, that's what we're looking for. I would be curious to know the answer if $M_\mathcal G$ is guaranteed to stop on any $G$ in some fixed amount of time in $|G|$. I'm also interested in any related results, where $\mathcal G$ is proved to be minor-closed with some other certificate (like in case of $TFNP$ or WRONG PROOF).
Update: The first version of my question turned out to be quite easy, based on the ideas of Marzio and Kimpel, consider the following construction. $M_\mathcal G$ accepts a graph on $n$ vertices if and only if $M$ does not halt in $n$ steps. This is minor closed and the running time depends only on $|G|$.