Does learning about the Lambda calculus have any take-aways for programming in modern languages, or is it more of a cool toy/mostly of interest for language designers and academics?
The lambda calculus teaches you how to build computation from a very simple evaluation rule called "beta reduction" which is just a fancy name for "search and replace" with function parameters. If you can find a system that can rewrite terms in such a fashion, you can compute whatever you want using the tricks you learn in lambda calculus.
For example, C++ templates were not meant to be Turing-complete, but they work by rewriting terms [1], so you can encode lambda calculus in it, and run arbitrary programs at compile time.
Philip Wadler does a better job than me at getting excited by lambda calculus. He calls it "the omniversal programming language", and claims that aliens will probably have discovered it [2].
I've found understanding the lambda calculus goes a long way to understanding functional programming—in particular, it does a wonderful job demonstrating just how simple the core ideas underlying languages like Haskell are, contrary to their reputations.
It's also a great way to get a deeper understanding of computation and how it relates to logic. The lambda calculus is special for being an incredibly minimal model of computation that's still expressive enough to write programs (albeit a little awkwardly). Expressing something you care about in lambda calculus is way easier than using a Turing machine directly or even programming in assembly!
I think lambda calculus encodes computable functions, but not algorithms, because changing evaluation order can make you go from O(n) to O(n^2) etc. To define time and space complexity, you need to assume something like Haskell's graph reduction. That's more complicated than starting with assembly and counting steps.
It gives a pretty facinating point of view onto computation in general.
You only the ability to define and call functions. Functions can only recombine their input. And somehow it is still as powerful as, say, python!
Don't need some built in math , invent numbers using function definitions. Don't need built in recursion, just implement it using functions as the y-combinator!
There are also some practical usages of the whole implement-data-as-functions idea. There are some libraries in haskell that do this, for instance, which means that the compiler can optimize dsl's.
I actually found myself factoring my code differently after learning lambda calc. It at the very least seemed to result in better code (c#) so I would say there is practical benefits.
Besides the uses of the lambda calculus for functional programming, Church Encoding is also a powerful tool for algorithmics, program architecture, and performance improvement.
