Mathematica has Infix [0], which expresses the adjacency with a ~ (because Mathematica reserves pure blankspace for multiplication). But it works fine to do eg. `"hello"~StringJoin~" world"`; I was always surprised we could only have the predefined operators in many other languages and we couldn't define our own.
This seems like a great attempt. I would be worried about how much parsing and backtracking might be required to infer the infix precedence in a totally general system (like garden-path sentences[1]) or actually ambiguous parse trees (which is cured by adopting some rule like right precedence and parens, but what rule you pick makes some 'natural language' constructions work over others).
Haskell supports user-defined operators (made up of symbols) and also lets you use functions in infix position by surrounding the name in backticks, e.g.
Similarly, Agda has a well-typed mixfix operator syntax – you define a function like (_foo_bar_baz) and can automatically write "X foo Y bar Z baz". It does mean that the operator parser has to be extensible at runtime, but it's not a huge cost for a dependently-typed language.
This seems like a great attempt. I would be worried about how much parsing and backtracking might be required to infer the infix precedence in a totally general system (like garden-path sentences[1]) or actually ambiguous parse trees (which is cured by adopting some rule like right precedence and parens, but what rule you pick makes some 'natural language' constructions work over others).
[0] https://reference.wolfram.com/language/ref/Infix.html
[1] https://en.wikipedia.org/wiki/Garden-path_sentence