Why is it so important to avoid the problem laid out by Banach-Tarski on the first go through the concept? We, as a species, did not even realize such a problem existed until about 1920, and yet we still managed to build stuff based on what we knew.
The history of mathematics is a history of ever-increasing standards for proofs. "Proofs" that, say, Euler wrote down that were perfectly valid for the standards of their time would probably not accepted in todays standard. I personally also think it's plausible that today's proof standard in "typical" math papers will not be acceptable in 100 years anymore, when they perhaps will additionally have to be computer-checkable (and computer-checked).
There clearly exist methods for solving problems that have nothing to do with mathematics. This should falsify your thesis.
A: Knives are for eating.
B: Eating is only a part of it. Knives are for handling food.
A: You can handle food with a fork. Your claim is false.
Specifically, "math is for solving problems" does not imply "problems are only solved by math".