> I could understand the basis for climate change if I took enough time out of work and family and did my research.
> With religious truths, on the other hand, investigations reach a dead end with "this is simply a matter of faith" or "you just have to feel this to believe it".
No religion worth its salt would ever present a proposition, point blank (e.g., the divinity of Christ[2]) and then claim you are required to believe it just because, or because "feelings". That is merely a caricature, a straw man, and if that were the case, then you could easily ask why this particular belief? Why not what the guys down the street are selling? New Age is big on feelings, a grand synthesis (read: hodge-podge) of superstition, so why not them? Why not the various kooky ideologies springing up in our times, nearly all of which follow the same gnosticizing script, with the added aggression and bullying they deploy when demanding you believe their claims simply because they make them. Certainly, there are some traditionally religious people who say the kinds of things you say about faith, today perhaps especially in the US, but it is neither wise nor charitable to judge a faith by the flaws of its weakest professors. And not only that, but not all faiths are the same. Not all faiths are actually faith, that is, rational faith.
Now, because faith is a kind of trust, a kind of uncertainty does enter the picture. But legitimate trust is not irrational, and so genuine faith is not irrational, and indeed, all men have faith; your distinction is not as profound as you may think, though certainly there are distinctions to be made and a certain diversity of methods ought to be considered (historical vs. empirical, for instance). Methodology is an important thing to bear in mind, specifically when recognizing, that rationality and modern, empirical science are not identical. This is the error of scientism, one easily refuted by the observation, that the claim that equates the two is not itself a scientific claim.
John Newman[0], for example talks about how the aforementioned assent can take place, and here is an article[1] that gives a brief summary. But a look at, say, the robust tradition of Catholic apologetic scholarship will certainly show, that this faith not only respects reason, but in doing so, proceeds from it. It goes where reason leads and takes off where human rationality is too feeble or circumstantially constrained to continue. There must be warrant, and reason and authentic faith cannot contradict. So here, too, you could, in principle, also do you own apologetic and theological and philosophical research and come to certain conclusions, just as you could, in principle, if not in practice, do your own empirical or historical research or whatever.
[0] https://www.newmanreader.org/works/grammar/index.html
[1] https://www.newadvent.org/cathen/05752c.htm
[2] https://www.peterkreeft.com/topics/christ-divinity.htm