Has there ever been a more-or-less successful attempt to write a truly modern math book, say, on analysis or geometry, one rooted in the more abstract mathematics of the last century, that is introductory in nature, i.e. that does not assume that a person has already been through a four-semester calculus sequence? Maybe something that anchors or connects it to some mainly modern subject, like abstract algebra or topology? Or that ties it more to foundational concepts, as in systems of relationships between concepts and fields, rather than, e.g., set theory. (I describe these, somewhat unconventionally, as foundational, because I think they are closer to the heart of of the practice of working mathematicians. Somewhat in the flavor of catagory theory, maybe?)
I have been feeling a need to review my college calculus, and it occurred to me that I might like to learn something new at the same time, maybe something that would give me a deeper appreciation of the subject (though Apostel was pretty deep). But since my original motive was that I have forgotten much of this over the last 35 years or so, I’d rather not focus on a book that assumes that I have all of it readily at hand. Also, I’ve been working my way through Pinter‘s A Book of Abstract Agebra 2nd ed., and it’s loads of fun. It is really nice to find that I can still do proofs, after not doing any for thirty years or more.
It sounds like you might simply want to look at an advanced calculus book which starts from an axiomatic point of view. I like Fitzpatrick's "Advanced Calculus" which starts from the axioms for the real numbers, and Tao's "Analysis I" which starts from the Peano axioms for the natural numbers.
But I don't know of any such book which assumes that the reader has never seen calculus. You could simply pair an advanced calculus book with a standard undergraduate calculus textbook, dipping into the latter as needed whenever the more advanced book moves into a new topic. You'll certainly find many more elementary exercises in a standard textbook, which might be useful.