I have seen many users here asking questions about problems in what they call "Spivak's Calculus Book". I have never seen the book, and information online is scarce. From what I've gathered, it is just a more rigorous calculus 1-3 book with harder problems. I have already taken calculus, and I am about to take analysis. Is there any point for me to buy Spivak's book, or is reading analysis books better at this point? Is this calculus book better than a standard analysis book? Why don't students who are gifted enough just start off with an analysis book instead of Spivak's calculus book?
It appears that there is tradition of studying calculus twice: first the non-rigorous version and then the rigorous version. Luckily I happened to study both at the same time (with school textbook being non-rigorous and Hardy's Pure Mathematics at home) and I did not find any benefit at all of the non-rigorous version.
I think the non-rigorous version is available because calculus is used in many applications and there is no need of a rigorous approach to learn the applications of calculus. Perhaps those who don't have to make a career in mathematics (or to use Hardy's phrase "students whose interests are not primarily mathematical") don't really need to study the rigorous version unless they would want to do it just for the sheer pleasure rigor adds.
But on the other hand if you have interests which are primarily mathematical it is a must to study calculus with rigor and the sooner you do it the better. I have myself studied Spivak's book but at a much later stage and apart from few nice problems (BTW all these nice problems of Spivak's book are available with solutions on MSE) I did not find much which was new compared to Hardy's book. But if you have never studied any books which treat calculus rigorously then Spivak's book is a very good option.
Your question "why not start off with an analysis book instead of Spivak's calculus book" appears to make sense on the assumption that "calculus" and "analysis" are on similar level the difference being lack or presence of rigor. This assumption is not entirely true. In most cases people are introduced to calculus at young age (high school age 16 years or similar) and the subject is treated in a concrete manner whereas most of the analysis courses (there are many real analysis, complex analysis, functional analysis and perhaps many more which I am not aware of) focus more on abstractions and some complicated and difficult topics like Lebesgue Integration, Fourier Series which are not covered in calculus courses.
In my opinion it is better to learn the usual calculus stuff (limits, continuity, derivative, integral) with full rigor and study advanced topics in analysis courses. Unfortunately many real-analysis books do cover these same calculus topics with rigorous proofs and then jump on to advanced / abstract topics. I personally don't like this redundancy and even the attitude that you can sacrifice rigor while studying/teaching mathematics.
I personally think that Spivak is a perfect book to read just after calculus and to get ready for real analysis. Spivak's book will teach you the basic tools of analysis, and previous knowledge of calculus will help greatly to follow the book.
And, as mentioned in the comments above, the book is wonderfully written and a pleasure to follow. It has interesting topics and challenging exercises. If you have the time, I think that going through this book will be a good investment.