Unless you had hit upon a very magical binary function where certain special values of the second parameter happens to coincide with useful unary functions, without those values trampling on a useful binary mode or region of your binary function, but the search space for such a special binary function is so large that you shouldn't demand us to disprove the existence, but rather employ your non-surprisal at the EML result and challenge you to present such a binary function, so we can challenge you to demonstrate how it captures binary functions like addition,products, exponentiation with arbitrary base etc.
So, can we see your construction, or if you refuse to present one, we may conclude you have implicitly reconsidered your position and understand the theoretical elegance this EML (and presumably many other) basis brings?
This requires expressing binary functions, like addition and multiplication.
You cannot do this by using only the set of unary functions, which can indeed be generated by a function with 2 parameters, one of which selects an unary function.