We also used Newtonian dynamics to get to the Moon, notwithstanding that Newtonian dynamics was known at the time to be wrong! (I assume the computers then weren't powerful enough to do the relativistic calculations, and it wasn't necessary from a practical standpoint anyway.)
As Alamo-Girl will tell you, having read Popper, no number of confirmations or verifications of a theory will "prove" the theory -- that is establish it to be necessarily true. It is always possible that there is some other theory (possibly one that no one has thought up yet) which will will pass all those same verifications, as well as additional tests that the existing theory will fail to pass. What's more, that new, more successful theory has exactly the same status. It too stands to be falsified, or surpassed by some better theory.
Even if we did manage to formulate some theory that was completely true, there is no way we could know that.
We can "prove" things in mathematics and geometry (establish them to be necessarily true) precisely because we (humans) establish or define all the formalisms of the system, or because we know all the rules of the system, or because they follow the laws of logic, or the very same rules of inference that are used in formulating the proof.
We can only "prove" scientific theories if we know in advance all the formalisms of nature, but of course we do not know this. It is precisely to discover and investigate the formalisms or rules of nature that we create theories.
While Popper may have said some interesting things about science, I do not consider him the final answer on everything. While I like Popper's political views, I do not like or agree with his overall skepticism. To say that we really cannot know anything is silly. If we do not 'know' anything we would not be able to function. To me it is a denial of life. Even Hume, who really wanted to deny that anything was provable, did not dare go so far as to deny that science could not prove anything.
Further, I am not talking about theories, I am talking about facts. Scientists do not sit around thinking up stuff. They spend their time proving or disproving things by experimentation, observation, and any other way which can help them determine the truth of a situation. They often spend a large part of their lives trying to prove how something works. The Law of Gravity has not been disproven, the atom bomb proves E=Mc2, and there are thousands proofs out there to scientific statements. Are the theories behind bridge building fairy tales? If you think so then perhaps you should not ever cross a bridge. After all who wants to risk their life on a fairy tale?
Indeed, the bar for "proof" is high beyond reach. Even in relativity and quantum mechanics, reference is made to accuracy within certain limits and the word "proof" does not appear.
The most reliable fundamental physical constants are also shown to certain limits: NIST
However, to most people, the distinction probably doesn't make much difference since they can and do rely on the most current thinking in various matters.