Being able to do multiple analysis of the same data, and find one where p < 0.05 has revolutionized academia.
Entire careers have been built around p-hacking.
Of all the statistical idea, it is the one which arguably has had the most influence in the life of the academician.
Yeah, the term only came up recently. But ultimately p-hacking is "just" exploiting freedom in the possible interpretation of statistical results. Also it wasn't really "invented", it's just what humans do if they have data in front of them and some expectation what the data should tell them. I'm pretty sure the first p-hacker was the first person to do any form of early statistics.
I guess the greatest invention would be a method to prevent p-hacking. (And yeah, I know about preregistration. It doesn't work the way it's done right now, because a) most people don't do it, b) preregistration is rarely precise enough to really prevent p-hacking and c) nobody cares if you switch your outcomes.)
It's much much harder to lie with effect sizes. For instance, in genetics there are entire careers built on an association of certain genetic polymorphism with a particular phenotype. Even if the p-value can be reproduced with huge sample sizes, the effect size of the association is usually tiny.
In other words, if you carry the risk allele the odds of developing a disease are usually only slightly bigger than if you didn't. But this is often not told to funders! Focusing on effect sizes would drive research effort to things that actually make a difference.
You can do effect sizes with something as simple as a confidence interval, or with a fully fledged Bayesian model that tries to estimate the posterior distribution of some parameter.
This really should be the standard for any meaningful science.
That said, you can do something called multiple comparisons corrections, where you take into account the fact that you're doing multiple tests in the same go. In this case you'll end up being more conservative, and it's not the false positive rate you'll be controlling - instead you'll control the family-wise error rate, which basically has to do with whether _any_ of your positive results are false positives.
Submiting a large number of individual registrations for a single set of data is just not the norm and so it would be suspicious to begin with. (Okay, I will admit that on the order of 2-3 registrations for very different categories of study may not be as strange; but certainly 20 would signal something very odd).
While impractical in many cases, I think the gold standard is to approach a problem from as many angles as possible, and relate it to disparate sources of knowledge, including theory.
For example, the accepted values of the physical constants are established by munging the results from a wide variety of different kinds of experiments, guided by theory. They don't just repeat the same experiment over and over. These constants are robust enough that you can hang your hat on them for virtually any practical use.