I don't think it's possible to divorce "proof" and "application" as cleanly as you'd like.

For instance, let's take a very common "application" of statistic: given a test that has false positive probability p, you can apply the test n times, and the false positive rate of the composite test is p^n. (For e.g. this is used to analyze bloom filters, or the Miller–Rabin primality test, or hash tables). However this is only true if the tests are independent. If you have to apply this result to analyzing (for e.g.) some new data structure you wrote, you'd first have to prove that the tests are independent. And maybe it's just me, but I find that if I use heuristics to check if some events are independent, I really often get it wrong.

Another example: the uniform limit theorem (https://en.wikipedia.org/wiki/Uniform_limit_theorem) is quite useful, but to properly apply it, you have to understand the difference between uniform convergence and pointwise convergence, and maybe it's just me, but I found the difference very unintuitive when I first encountered them (in a standard proofs-based analysis class). Even now, if you gave me some random series of functions, I can't really imagine using heuristics to check it uniformly converges to its pointwise limit, I'd want to try to write down a proof to be sure. So this useful tool is gated behind understanding some (to me) subtle proofs.

Haha, you were so careful and still got the classic "That's not serious mathematics!" response.

You might want something less applied, but I highly recommend Burden & Faires "Numerical Analysis" and Trefethen "Numerical Linear Algebra". The various interpretations provided by Trefethen for understanding a matrix, eigenvectors, singular value decompositions, etc. are amongst the most insightful descriptions I've come across.

What you're describing as "real math" is no more "real" than the kind of "math" you do claim to want. They're both math. What you think of as math for math's sake is not necessarily any more for math's sake than for the sake of something else. As other commenters have tried to explain, the two go hand in hand often enough anyway. (And you're right they sometimes don't, go to some random math journal and pick a random paper and it'll probably be something neither you nor I can hope to understand any time soon, with application seemingly nil in any way we could see. A lot of high level math is like that. Maybe some of it is best termed 'exploration'. Nevertheless, it has no claim to "real" math, and I don't think those guys are responsible for US math curriculum.)

Drilling proofs is a valid kind of drilling and can be an effective way to learn something. Not necessarily the only way, sure. But there's nothing fundamentally different or "more real" about drilling proofs vs drilling grade school multiplication problems. You'll memorize things, you'll see patterns, develop heuristics, gain intuition.

Application can sometimes be tricky; did those grade school multiplication drills have application? Are they granted more application by phrasing things in terms of word problems around counting apples or whatever rather than the compressed a times b = blank expression? Well, sometimes the application of proofs will be more direct, sometimes less, and can be phrased better or worse, more realistically (and necessarily more complexly) or less so, like any other exercise, whether it wants a proof or not. CS proofs about big-O complexity are applicable to analysis of algorithms, which is pretty important if that's your focus. Though most problems you could find to drill specifically on big-O (as opposed to other parts of algorithm analysis, like recurrence relations) would likely take the form "find the complexity of this" or "given the complexity is such, estimate..." rather than "prove that...". There are many things no one knows how to prove that are still an area of study, clearly proofs aren't the be-all-end-all. Anyway, the mental processes involved between something like "find x, the hypotenuse of the triangle" and "prove the Pythagorean theorem" often aren't that different. There are multiple ways to prove it, you could drill on them.

And technically, computer programs themselves can be thought of as proofs (Curry-Howard correspondence) so if you've ever written a program that terminates you've written a proof... Proofs don't necessarily have the form or flow "by axiom 1, axiom 2, theorem 34, modus ponens on this, proof by contraposition on that which we'll name lemma 8, and by induction over the integers here, we have proved blah, QED".

And if you grant simple algebraic symbol manipulation as something you would do to solve a word problem, well, that itself is a style of proof. (There's a whole automated proof engine written entirely on the basis of substitution, the same process you use in a simple algebra problem of substituting x + 3 = 10 with x + 3 - 3 = 10 - 3 and reducing to x = 7.)

But fine, no proofs, not even in disguise! What is it that science, engineering, and technology focused subject books that use math only as needed without bothering to prove things when unnecessary (some having exercises and drills of word problems from realistic circumstances) don't do to solve your craving?