Once you guarantee that the function takes the specific form, you can manipulate it in ways you can't manipulate the general-case function, and it turns out tho

Once you guarantee that the function takes the specific form, you can manipulate it in ways you can't manipulate the general-case function, and it turns out those manipulations give you enough power to define almost anything else you'd want to do. In λ calculus, the only way one term can inspect another is by applying it. You don't get an intensional view of a λ abstraction. The other language features that get built on top of this (conditionals, tuples, etc.) only rely on functions' extensional features.