What you are referring to is more 'scientism' than the 'scientific method'. Scientism has the veneer of the scientific method, but is completely divorced from it and is riddled with politics, corruption, studies that cannot be replicated (or thus falsified), and so forth.
Most of what is called the scientific method is scientism in the modern day. An easy example is asserting that men can get pregnant, that surgically cutting off your genitals and injecting chemotherapy will change your biological sex and/or chromosomes, or the completely rushed COVID-19 vaccinations which did not pass any sort of established human testing standards (shit takes decades, like measles and all prior vaccinations) besides openly changing the long-standing definition of vaccinations. More subtle examples are the widespread falsification of studies for political reasons (global warming comes to mind) or simply to not get kicked out of academia as a professor due to not churning out discoveries on a linear timeline.
Returning back to your original question, the 'Republican' version of this is ... well, the scientific method.
My personal insight as a trained scientist and mathematician is that truly following the scientific method is to accept that there's always undiscovered or currently unrecognized ways of knowing things are true outside of the scientific method. By analogy, things like animals having genetic intuition to do things would fall under this. If there are beings in the currently verified at least 5 dimensions beyond the ones we actively experience, it would make sense that knowledge could very likely be passed down dimensions to us mortal beings which would also sidestep the scientific method as a second example. As a third example, arguably, we could say that empirical forms of knowledge such as knowing the Pythagorean theorem is true by fitting a bunch of triangles and squares or objects together without any written or symbolic knowledge nor knowing why they are true might fall under this, as pre-historical civilizations certainly did.