The example you've described (radio signals) is just a method of winning an argument. It doesn't actually help you, you practically do it by writing a computer program that tries out different parameters to get the right one.
Let's say you need a communications channel with a 300 Mbit/s capacity. Shannon's theorem lets you know what kind of parameters make that possible -- namely, bandwidth and signal-to-noise ratio. From there, I can make an informed decision about the entire signal chain. Once I choose the bandwidth, the theorem tells me the SNR so I can assign a noise budget to each of the components. If we assume that the noise is Gaussian, we can calculate total noise level as the root sum of squares of the noise levels of the individual components (don't forget to multiply by the gain). I can look at an amplifier IC's spec sheet and immediately say, "that's too noisy" or "that's overdesigned, I think I can get something cheaper".
And thanks to theorems we got from the field of real analysis and statistics, we know that the sum of Gaussians is itself Gaussian. So rather than sticking two components together and measuring the noise, or running a computer simulation, I can simply square, sum, and square root.
Each of these theorems reduces the amount of work necessary -- whether by pen and paper or by computer -- by an enormous factor. But if you can describe in similar detail how the computer program would work, I'll acknowledge your greatness.
Really? What happens when you need to do a similar calculation across hundreds of millions of data points?
And you can't just "write a computer program that tries different parameters". You have to prove that your numerical method solves a certain class of equations first, otherwise your rocket will fly sideways, if at all.
I assume that all those automatic methods you mention are passed down from above in some sort of holy scriptures that we're supposed to blindly believe and use? Or maybe some "pencil pushing" mathematician came up with them first and _proved_ that they actually work?