I think that you’re right that most people who employ the argument don’t have a firm grasp on statistics, and I’ll put myself in that category. I recently read an article about dream premonition written by (I believe) an agnostic, essentially saying that just because you dream about someone dying and then they do, you’re not really experiencing anything outside of what is statistically likely. Billions of dreams are had a night, a great many of them probably involve the death of loved ones, and everyone dies eventually, so the odds are actually really in favor of that sort of thing happening, even if dreams are governed only by blind chance. We just think that this is unusual because we don’t focus much on the dreams that don’t come true - what they call confirmation bias.
Dawkins, if I’m not mistaken, applies that argument to biological evolution, and utilizes natural selection, saying basically: evolution necessarily involves taking trillions of chances (each unique DNA code offers a new chance at an evolved characteristic, or at least a new combination of existing traits), and natural selection favors the beneficial outcomes, so it’s not that random. For example, if you evolve a dominant-gene genetic disorder which kills you by the age of 3, that’s not getting passed on - it’s an evolutionary dead-end; but if you evolve a trait which lets you live longer and/or reproduce more, that is getting passed on, so natural selection favors good characteristics (hence, the world we see today, with complex organisms with well-functioning design). Meanwhile, we ignore the badly-functioning genes - the infant genetic disorders (like Tay-Sachs) and so forth, either because of confirmation bias, or because those animals who suffer from the bad genes are less visible (since natural selection disfavors them).
I don’t know that I would disagree with that assessment of evolution very much, except to say that I think the Hand of God is at work in it all. I think He created natural selection, the laws of the universe, and so forth. But I think that the science is roughly there. Catholics believe in sacraments - God working through the everyday material things in life, like water, wine, bread, and even sex and marriage. If He can create new life using the existing DNA of parents, why shy away from macroevolution’s claim that He made various species from preexisting species? At the least, I think the Church is fine with that. We would reject, of course, any attempt to say that since science is at work, God must not be. It’s a logical fallacy, like saying that if you can measure the speed at which a domino falls (or even prove that its fall was caused by the domino before it) that it means that no human involvement started that reaction.
Where I think the improbable nature of the universe comes into play is with the non-biological universe: in other words, from the Big Bang to the first organism. Natural selection can’t explain things, and unlike natural selection where billions of organisms give you plenty of try-and-fail time, the non-biological universe was a one-shot chance. Had the gravitational constant been even slightly off, or the rate at which the universe cooled, or had the strong and weak nuclear forces been somehow different, not only would we not be here, but the universe would have imploded and destroyed itself. Here I think the incredible odds are very important: everything from the first few nanoseconds of the Big Bang to the creation of our solar system and the unique way in which Earth just happened to be in a perfect climate to sustain life defies the odds. (This is especially true the closer to the universe’s origin-point you get - there are probably billions of planets which could potentially have shared in Earth’s luck, but only one universe).
Scientists have been forced to design the Anthropic Principle and the Multiverse Theory to try and account for the fact that blind statistics can’t get you (a) from a lack of matter to matter; (b) from the presence of matter in a time-stasis (which is what the Big Bang would be - Augustinian time) to a Big Bang; (c) from the Big Bang to a second after the Big Bang; and (d) from the Big Bang to a universe sustaining life in any form or capacity.