What is the contribution of lambda calculus to the field of theory of computation?

I'm just reading up on lambda calculus to "get to know it". I see it as an alternate form of computation as opposed to the Turing Machine. It's an interesting way of doing things with functions/reductions (crudely speaking). Some questions keep nagging at me though: What's the point of lambda calculus? Why go through all these functions/reductions? What is the purpose? As a result I'm left to wonder: What exactly did lambda calculus do to advance the theory of CS? What were it's contributions that would allow me to have an "aha" moment of understanding the need for its existence? Why is lambda calculus not covered in texts on automata theory? The common route is to go through various automata, grammars, Turing Machines and complexity classes. Lambda calculus is only included in the syllabus for SICP style courses (perhaps not?). But I've rarely seen it be a part of the core curriculum of CS. Does this imply it's not all that valuable? Maybe not and I maybe missing something here? I'm aware that functional programming languages are based on lambda calculus but I'm not considering that as a valid contribution, since it was created much before we had programming languages. So, really what is the point of knowing/understanding lambda calculus, w.r.t. its applications/contributions to theory?