A lot of the strength in cryptography is lost in areas that depend on trust. Like trusting that the vendor doesn't put a backdoor in your system, or trusting your OS doesn't break your firewall, or that any third-party CA's are actually trustworthy, or there isn't a weird compiler bug that kills your entire encryption system. These things may be tested against and prevented one-by-one, but they are overhead, which makes the notion of security a matter of risk management. Cryptography tries hard to reduce the reliance on trust, but it's always a big issue.
More of a theoretician, I was once lectured by a competent software engineer that in security the devil is in the details, where any system has to stand the test of time and often goes through many fixes to be just usable. Theoretical security guarantees are much stronger what is often realized in practical security systems, because implementation details fall through cracks that are covered up by theoretical abstractions that breed high-level cryptography.
US isn't seeing any of the benefits (yet) because it's a massive power that can borrow and print money willy-nilly, and the currency is mostly under control at the moment. But not all countries are like that. There are countries where something like Bitcoin can be a viable alternative to official currency, like Argentina. Bitcoin is flawed, but it guards against certain attacks that can be performed to a currency, in a way that is sometimes useful.
On the other hand, the way Dell accepts Bitcoin doesn't bear a lot of risk; a third-party accepts Bitcoin for them and pay them in cash, and absorbs fluctuation risk in return for some fees. It relies on customers bearing the risks themselves. It isn't a big step for Bitcoin; it is only a very small step towards forming a sufficient number of nodes of a skeleton that may or may not turn into an actual ecosystem that eventually circulates Bitcoin. But isn't a terrible decision for Dell either.
Even "artificial" methods are "discoveries"; "this method can be used to achieve that" is simply a consequence of the laws of nature, discovered by the inventors. An elevator may be seen as man-made, but the engineering of it is still a method, whose usefulness is merely a consequence of laws of nature, hence "discovered". Yet a patent does not "protect" a single elevator, but the method that makes it work. A star can't be patented, but neither can an elevator.
What you advocate is also against the spirit of patents, which is to protect science progress by providing financial incentives to make effort to contribute to science, whether or not some people would think such contribution should be be labelled a "discovery" or an "invention".
I don't have an answer as to whether we should patent mathematical methods, but your argument against it is very weak.