It's an interesting question. In general, I don't think people are more credulous of science than they ever have been. The War Room is a terrible sample because it's filled with weird extremists for the most part (myself being one of them in many ways). But I think you should distinguish between faith in science and faith in the scientific establishment. I don't think most people have much faith in science per se because as you correctly point out they know very little of it. Most scientists know relatively little outside their specialty, there's just too much to know to go deeply into more than 1-2 areas. What I think people have faith in is the scientific consensus and the scientific method, even if they wouldn't express it like that. We believe it when scientists tell us stuff because science has done more to explain the natural world, make testable predictions that turn out to be true, and better human life than any other institution in human history. There is a ton of bad science out there, but most of it falls by the wayside in time because that's how science proceeds.
You mention things like competing models, inconsistent evidence, etc. as reasons people shouldn't have faith in the conclusions of science. You're only partially right. People shouldn't assume that the scientific consensus is true in some absolute sense, they do so because they don't understand the nature of 'truth' in science. It's always contingent on either new evidence or a new theory that explains the existing evidence better and more parsimoniously. People love to site Popper and the idea of falsifiability, and while that's an important aspect to some sorts of science the way science really proceeds I think is closer to Lakatos's view of theories in line with the evidence, tinkered with in areas they're found lacking, retained only until a better or more complete theory comes along at which point they're discarded. Science is not a collection of facts it's essentially a set of working conjectures. That said, it's the most effective set of working conjectures for explaining the world we've ever seen and as such peoples' faith in the curators of that body of knowledge is wholly justified.
So while I agree with you that people should be more aware of the epistemic uncertainty inherent in science and not worship scientists or the scientific consensus, in general the faith people in have in science to explain the world is completely correct.
