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).