Edge-disjoint paths in trees
Here are some ideas:

Suppose that $\ell$ is a leaf of $T$ which doesn't belong to $S$. Then we can safely remove $\ell$.

Suppose that $\ell$ is a leaf of $T$ which does belong to $S$. If its unique neighbor is also in $S$, then we can safely pair them up.

Suppose that $\ell$ is a leaf of $T$ which belongs to $S$, but its unique neighbor $r$ doesn't belong to $S$. We can safely remove $\ell$ and replace it in $S$ by $r$.


(How to implement these ideas efficiently is a different matter, but for an existence proof these ideas suffice.)