The question by curiousdannii and the extensive answer by lemontree here address the basic workings, advantages, and disadvantages of type theory. I have a related question in this area. As I understand type theory and as described by lemontree, type theory necessarily construes syntactic structures as being generated in terms of one input and one output. This suggests that the syntactic structures generated necessitate binary branching; ternary branching does not seem possible, at least not in the illustrations of type theory that I have encountered.
My question in this regard concerns the potential that this aspect of type theory can be chalked up to the disadvantages. For me, there is substantial empirical evidence in favor of n-ary branching in syntactic structures. If type theory is incompatible with this evidence, then what good is it? It results in a semantics that is incompatible with the syntax.
If there are versions of type theory that are compatible with n-ary branching, then what are they, and how widespread and accepted are those versions?