
Weinberg's 1972 work, in his description, had two purposes. The first was practical to bring together and assess the wealth of data provided over the previous decade while realizing that newer data would come in even as the book was being printed. He hoped the comprehensive picture would prepare the reader and himself to that new data as it emerged. The second was to produce a textbook about general relativity in which geometric ideas were not given a starring role for (in his words) too great an emphasis on geometry can only obscure the deep connections between gravitation and the rest of physics.
Padmanabhan: Gravitation: Foundations and Frontiers - A modern point of view
Feynman: Lectures on Gravitation - Feynmans unique insights are a rare gem
Mukhanov: Introduction to Quantum Effects in Gravity - The best introduction to quantum gravity (and also quantum field theory)
(It’s not like the subject experts don’t realize the dangers of one or two observations or approaches holding up a theory, especially in a field like astronomy where you’re stuck with the experiments nature deigned to perform for you. That’s why the Bullet Cluster is not the knock-out argument for dark matter that it appears to be—it’s a single object, maybe it’s some kind of freak occurrence we can’t really sort out from here.)
Generally speaking, Weinberg’s Cosmology (not to be confused with his GR textbook Gravitation and cosmology from the 70s, the cosmology parts of which are much too old to be useful) is a bit peculiar in a way that might be helpful here: Weinberg is undeniably one of the giants of modern physics, but not in cosmology, astronomy, gravitation, or similar—he’s a high-energy person first and foremost, with connections to other field-theoretic pursuits. So AFAIU writing a cosmology textbook was just his way of learning how the subject developed over the three decades when he wasn’t looking (spoiler alert: a lot).
Thus he tends to pay attention to which pieces of reasoning and evidence hold up the subject to an extent that people immersed in it don’t—including things that a normal astronomer considers elementary knowledge not really part of cosmology proper, like Hertzsprung—Russell (and, to be fair to the astronomer, it really isn’t).
That doesn’t mean that he’ll hold your hand: if you don’t know special relativity, the book’s probably going to be mostly unreadable (Taylor—Wheeler and a bit of Lightman—Press—Price—Teukolsky be with you); if you don’t know enough Riemann geometry to write down a geodesic equation, ditto (try Misner—Thorne—Wheeler maybe? it really is a cursed subject); same for not being able to derive said equation from an action, or not knowing how blackbody radiation works. All of those are proven as well as anything can be, though, and hold up much more than just cosmology.