Neither lambda calculus, nor Turing machines "model computation on" recursive functions. Rather, these two, and all the other models of computation are equivalent as they compute the same class of functions from the naturals to the naturals. And we call that class the class of recursive functions.

That being said, a computer is a physical device that is able to compute this entire class of recursive functions. It is what makes it a computer and not a pen or a chair. It's not some esoteric notion. It's the whole thing about it.