Several of the papers citing Stallings paper A topological proof of Grushko's theorem on free products are using the binding tie argument. Here are the ones I found using a Google search. More can probably be found using MathSciNet.
- Jaco [1968] Constructing 3-manifolds from group homomorphisms
- Heil [1972] On Kneser’s conjecture for bounded 3-manifolds
- Feustel [1972] A splitting theorem for closed orientable 3-manifolds
- Bowditch [1999] Connectedness properties of limit sets
- Bellettini, Paolini, Wang [2021] A complete invariant for closed surfaces in the three-sphere
Several searches failed to find the precise statement you mention. However, Lemma 3.2 in Jaco's paper comes close (he maps a surface to a graph, pulls back midpoints, and so on).