I question some of the more specific claims here. Analysis isn't the "why" of calculus; you could even argue that it's a more precise form of how to do calculus, along with a bunch of other things... Saying that pure math books rarely contain the "hows" is a strange slant on what a "how" is. You could even turn things around and say that "why"s are reasons we do something, i.e. the applications of pure math concepts, but that's just semantics at that point.
Without knowing your motivation, I find that it's common for students to feel they have FOMO ("fear of missing out") on some aspect of math that their education seemingly failed to provide. I have seen precalculus students who want to see proofs for everything, when it's pretty unlikely that that would enrich the precalculus experience for most students.
I will borrow an example of what rigor does in general. It doesn't make the standard cases more meaningful. It does clarify the exact boundaries of what works and what doesn't work. For example, if we are dealing with infinite conditionally convergent series, we can get into some trouble if we don't understand the problems with trying to sum that series. But in most walks of life, you can go weeks if not years without ever having to deal with that eventuality. (Or, you know, encountering a function that is continuous everywhere but differentiable nowhere.) Most of rigorous math is like that. If you care about it, it matters, but if you don't, then it probably doesn't. (It's more like chess strategy or literary analysis, as opposed to traffic laws or personal hygiene, which are things you can't safely ignore in most cases.)