Many of the papers citing Stallings paper A topological proof of Grushko's theorem on free products are using the binding tie argument. The ones I found are the following.

1. Heil [1972] On Kneser’s conjecture for bounded 3-manifolds
2. Feustel [1972] A splitting theorem for closed orientable 3-manifolds
3. Bowditch [1999] Connectedness properties of limit sets
4. Bellettini, Paolini, Wang [2021] A complete invariant for closed surfaces in the three-sphere

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.

1. Jaco [1968] Constructing 3-manifolds from group homomorphisms
2. Heil [1972] On Kneser’s conjecture for bounded 3-manifolds
3. Feustel [1972] A splitting theorem for closed orientable 3-manifolds
4. Bowditch [1999] Connectedness properties of limit sets
5. 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).