My argument is that basically all of what Sagan says is wrong.

The argument from authority is paramount in science; in fact it is the entire basis of the scientific enterprise. This is why it is important for you to ridicule other ways of knowing, as you did above ("You'll have to use the scientific method to research whether your alternative methods result in better outcomes for patients. Otherwise its just faith or some other nonsense."), implying that there are no other valid or useful epistemologies besides science. Of course this is not true, because the vast majority of people are not scientists, and yet are able to gain useful knowledge about the world without making use of "scientific methods".

You have continually asserted that all it takes to reject incorrect scientific knowledge is for one person to show up with new observations controverting a theory. But this is not true; it is not sufficient for you to do an experiment showing X to be false or Y to be true - because science produces knowledge by consensus. The basic requirement of science, above all else, is that your result must be peer-reviewed. That is it; there are no requirements of method, no statistical thresholds that must be crossed, no rules about the kinds of evidence that are admissible. The entry of your experiment into the domain of science requires you to cross this threshold. Even after that your experiment faces further review and scrutiny from members of your community.

This immediately means there is an important social dimension to doing science, based on a principle of authority: who gets to do peer review? The answer is: those who have been accorded respect by the community of scientists.

What this means is that powerful scientists are gatekeepers on what is considered to be "true" according to science. It is their judgement that determines what is scientifically valid and invalid, above and beyond the evidence. In general we expect that these people are correctly motivated and believe ideologically in the project of science, are good moral actors, scrupulous, etc. - but all of these things must be true in order for science to succeed.

Without this foundation of authority, science has no mechanism for validating evidence, which, unfortunately, is not self-evident. It is up to these people, and their judgment, to make the determination that some hypothetical is consistent or inconsistent with the facts - that is, whether something is true. It has been observed many times that this can hold up the progress of science.

And, to underscore, this is the extent of the scientific method. There are no rules for what the reviewer must do; they are merely handed a paper, told to critique it, and their determinations are handed back for further argumentation. The bounds of the disputation are up to the disputants. What this means is that the actual epistemology (how we decide "what is true") is continually shifting. One might summarize this by saying that the "scientific method" at any given moment consists of the set of instruments (physical or conceptual) that the community of scientists brings to bear on a problem. Right now there is a vogue of meta-analysis papers; meanwhile, multiple-test correction didn't even exist before 1960.

This is just a taste, I'm not really prepared to develop what should be a book-length argument in a HN comment, but it indicates basically what I am saying: there is a lot, lot more to the "scientific method" than falsificationism can present to us, and the sociological dimension of scientific knowledge is equally important to understand, but underappreciated by people like Popper.

Most of my thinking here has been shaped by two books, both seminal works: one is "Leviathan and the Air-Pump" by Steven Shapin and Simon Schaffer; the other is "Against Method" by Paul Feyerabend. I recommend you read them both.