I am aware that for applications (for many engineers for instance), the "early calculus" is all they really want and need to solve their practical problems. (No need to study notions such as uniform continuity, or view Calculus as theorems based on precise definitions).
However for an aspiring mathematician, could one begin with real analysis, and along the way, introduce the student to many examples? So they would see calculus along the way in the form of concrete examples of the theory, rather than studying calculus first for a number of years, and then learning real analysis separately.
It seems this is a more efficient approach, for a student intending to study mathematics further.
This may not be an answer in the strict sense of the word on this site (my apologies ... delete if you must). But, as an analogy, I ended up studying master's level point-set topology before I studied undergraduate analysis and it was the best decision I ever made.
In my analysis class, my peers were worrying about specific examples while I was thinking about the overreaching concepts. I had an edge.
However, “Replacing” Calculus with Real Analysis in the curriculum would mean, IMHO, attacking the problem of "mathematical maturity."
Mathematical maturity might be a problem. Most undergraduates spend a year, or so, studying calculus while they gather their logical foundations and learn about mathematical argument.
"Out of order" curricular decisions are a good thing at the right time.
So, I'll answer a question with a question:
How would you prepare a typical 17-year-old student for Analysis before calculus?