So there isn't really a single scientific method. That is a myth we tell students for pedagogical reason. It wasn't introduced or invented or anything like that. The development of what we today science was a fairly continuous process that stretches back thousands of years. There are lots of methods and lots of philosophies that have, over time, combined to be part of different forms of scientific practice (and different forms of science may or may not ascribe to any set of them).

What did happen, and this is germane to your question, is that in the 18th century, people for various reasons began proclaiming that science was an amazing new source of authority for knowledge and that it should be used to actively displace other forms of authority, notably religion and politics. This is largely what the same guys who decided that they were part of an "Enlightenment" were about, and this message got further amplified in the 19th-century as science professionalized and became something more established institutionally.

What you refer to as "trusting science" is what we sometimes call Scientism, which is not the practice of science per se, but a sort of ideological argument that says, in short, that science should be the ultimate authority on virtually all questions humans might have, and that the scientific community is the only community that can police the boundaries of what is and is not "science." Not all scientists advocate Scientism, and it is, again, distinct from the practice of science itself. There are also varieties of Scientism which are more or less "dogmatic" about it.

There isn't any distinct "starting point" for Scientism, though the Enlightenment is not a terrible place to start for the idea (one can certainly find earlier advocates, though; Sir Francis Bacon's idea of a scientific utopia, for example, is a particularly 16th century gloss on the idea, albeit one quite different from the more anti-religious aspects of the concept that would come later), and certainly during the Industrial Revolution it became extremely and explicitly prominent in the Anglophone world at the least. There have always been people, both inside and outside of the scientific community, who have pushed back on this particular approach to thinking about the role of science. I might add that there are many alternatives to characterizing society's relationship to scientific knowledge beyond "distrust science" and "trust science," and in fact it is probably hard to actually find people who explicitly avow either sort of extreme. Anyone with any sophistication in their thought would acknowledge, for example, that in many cases there is significant scientific uncertainty and thus it is not possible to simply "trust science" except inasmuch as it means to take science as ultimately the epistemic authority and not other sources.

Again, it is a red herring to ask whether Scientism is distinct from the "original idea" (there was no original idea), but it is a thing that has evolved alongside the development of science.