All of science is built on the principle of implication: truth can only imply truth. It cannot imply falsehood. False can imply anything, therefore you cannot rely upon it to derive truth. By starting with a set of truths, one can derive more truths without a falsehood sneaking in. This is logically necessary. If your system of reasoning does not provide this guarantee, then you have no system of reasoning at all, and no reliable way to derive new truths. You might as well resort to crystal balls and magic lamps.
If you derive a contradiction in a chain of reasoning, and the steps in the chain are all proper implications, then the starting premise cannot be true, because a contradiction is exactly the statement: T -> F (true implies false). The entire system of constructing proofs is a framework for ensuring that each step is indeed a proper implication that preserves the "truthiness" of the original premise. Thus, when you arrive at a step in which you imply false, you can collapse the chain of implications backwards by saying: "Therefore, step 9's premise is false. Since step 8 implies step 9, step 8's premise is false. Since step 7 implies step 8, step 7's premise is false." All the way back to: "Therefore, my starting premise is false."
If this entire mechanism has a hole in it, one that allows truth to derive false, then you have not discovered a powerful new truth: you have destroyed the very system that allows you to systematically derive truth to begin with. You have essentially destroyed your very ability to conduct science and mathematics at all.
If you wish to make any kind of scientific statement at all, via experimental or logical proofs, you need to have a system of logic which implements implication to begin with. And once you have material implication, you have non-contradiction and proof by contradiction as inevitable consequences of that. If you somehow lose proof by contradiction, then you also lose implication, and you now must find some other way to arrive at truth. But don't claim that your new system is based on logic, because it's not.
It might help to think of proofs like a sudoku puzzle. If you reach a point where the number 3 appears twice in the same row, you have a problem: you either made a mistake in your deductions, or you don't have a valid sudoku puzzle. What you cannot say is: "My deductions are all valid, this puzzle is a valid sudoku, and there are multiple 3s in this column. I have discovered that sudoku admits this possibility." It does not. The very definition of sudoku prevents this possibility. At no point will you have discovered that sudoku is flawed in a way that allows duplicate numbers.
That's because the definition of sudoku does not allow it. If you have a grid with duplicates, then it's not a sudoku puzzle. It's just a grid with numbers. And if you have a proof that leads to a contradiction, you don't have evidence that mathematics is broken, because the premise is true after all. What you have is a broken proof or a broken premise.
You are suggesting that Goedel's Incompleteness Theorems will save you by saying: "I proved that P leads to a contradiction, but Goedel tells me that P is really true after all!" That is not what the theorems say. They say that a sufficiently complex proof system is either incomplete or inconsistent. If it's inconsistent, then it's useless: you can prove T -> F, and thus you can prove any statement you like in it. You cannot use an inconsistent system to discover actual truth. Therefore, scientists and mathematicians choose to work in incomplete systems, with the consequence that some statements will be true, but you will not be able to prove they are true within that system (but you can by creating a larger system that contains it).
An undecidable statement is one in which assuming it true does not lead to a contradiction, but assuming it false also does not lead to a contradiction. And thus, the rules of the formal system are unable to determine whether the statement is true or false. On the other hand, if a statement derives false, then in that formal system, the statement is false, full stop.
There is clearly an asymmetry between truth and falsehood and how they propagate and what you can learn from them. And this is why these results can sometimes be confusing. But it is also why they are so powerful: truth is more restrictive than falsehood, which is why it is so difficult to maintain a web of lies. You only need one contradiction to see that the web is a fabrication. Truth networks need no self-consistency effort, because it is an intrinsic property of truth itself.
Systematically discovering truth works because material implication is a "truth-preserving" operation. Once you derive false (i.e., a contradiction), you can propagate that all the way back to the starting premise. If this process is not valid, then your entire system of procedure is also not valid, and you do not have a reliable system for discovering truth.