One of the reasons for infix notation in math is actually that it provides a 'visual' reminder of useful properties such as associativity, and possibly others e.g. commutativity or distributivity. If all we ever used was a strict LISP-like, function-based notation, such a reminder would be lost and understanding or manipulating non-trivial expressions would be quite a bit harder. The effort in OP is actually a way of generalizing this idea to broader settings, where one is dealing with something more complex than a single domain of number-like values, and a handful of operations on them. This is arguably how one should think of "graphical linear algebra" as well: the 'diagrams' one's dealing with there can be thought of as generalized expressions, so there's nothing overly strange in being able to manipulate those formally according to well-defined rules of some sort.