Reduction from Steiner tree to minimum set cover
It is your definition of Steiner tree problem that causes the difficulty here.

Your definition does not include the set of terminals $S\subseteq V(G)$. Then, a Steiner tree is a subtree $G'$ of the graph $G$ that spans $S$.

Keep this definition in mind, you can now move on to reduce $\mathrm{EXACT}$-$\mathrm{COVER}$-$\mathrm{BY}$-$3$-$\mathrm{SET}$ to your problem. The set of terminals would be the set of items to be covered, plus the root $r$. The construction is exactly your construction including your weight function.

But, the required weight $k$ is now $|U|/3$ where $U$ is the universe of items to be covered.



Clearly, a Steiner tree that spans $r$ and the whole universe should be a tree that can be rooted at $r$ with $k$ children as the set cover solution. Note that a Steiner tree is by no mean a rooted one but here we see that being seen as rooted at $r$ does help.