The computer industry wasn't built on Turing completeness. It's an evolution of practical technology starting with the invention of textile machinery (but you can also kind of argue it started with abacuses or even lines drawn in sand).
You are right. Evolution has its mysterious ways. But maybe it's only mysterious to us because our math has not evolved yet to a point where complex systems can also be properly described by theorems.
But then what is the use of the maths. We already have the wealth the comes from the technology. The maths doesn't add much (instead siphons off resources).
You're arguing for eating our seed corn. Yesterday's maths drove today's technology. Today's maths will drive tomorrow's technology, unless you have your way.
