I think you're confusing lambdas with lambda calculus. Lambda calculus is more than just anonymous functions.

To put it extremely simply, let's just say functional programming (the implementation of lambda calculus) is code with functions as data and without shared mutable state (or side effects).

The first one increases expressiveness tremendously, the second one increases safety and optimization. Of course, you don't need to write anonymous functions in a functional language if you don't want to.

As for why those "pseudo-functions" are useful, you're probably thinking of closures, which capture state from the context they are defined in. That is pretty useful. But it's not the whole reason lambda calculus exists.