This is somewhat of a hypothesis rather than a definitive answer, but one reason why vector calculus may no longer be the first proof based course at many colleges is because vector calculus involves teaching a lot of new material*. Real analysis, is mostly (at least in the first quarter/semester), material they have already seen before in calculus - minus all of the abstraction.
In math there are often two axes for the content being taught. (axis - 1) the abstractness and (axis - 2) the familiarity. The goal of introducing people to proofs is to venture down the "abstractness" axis that is rarely explored in high school and lower level college classes. To really focus on this abstraction it is best to do so with content that is familiar (i.e. 1-dimensional calculus). This way the students can really focus on the proof writing and the abstraction because they already have some intuitive (calculation based) understanding of the concepts from calculus. However, if you were to introduce rigorous proof writing in vector calculus you would have to teach a lot of new material along the way. This distracts from teaching the basics of proof writing; students can easily get overwhelmed if they have to do both at once. In addition when a student doesn't understand something, there is the added confusion of "do I not understand how to do the basic vector calculus calculations?" or "do I not understand the 'intuitive idea'?" or "do I not understand the proof" or all of the above. Often the struggling students can't even answer this question correctly, which makes it nearly impossible for them to overcome their difficulties without some major intervention (because they don't know what to work on).
I stole this idea from professor Steve Strogatz (the author of the somewhat recent math column in the NY times), from a talk he gave about what makes a "pop-math article" popular. He pretty much said, take something really familiar to the audience and dive into it a bit more abstractly, or talk about something unfamiliar but in the language of a concrete example that the audience can easily follow. If he wrote something that was both unfamiliar and abstract, he lost his audience. I think this general comment holds for students transitioning to higher math. Eventually, once they are comfortable with abstract thinking, you can then introduce to them new material abstractly, but not until they master proof writing.
My guess is that the real answer is going to vary a lot from school to school and have to do with how they sequence their other courses, but I feel like the above guess is a pretty good one. Note that there are also obvious pitfalls to this method, especially for advanced students, who really could benefit from going abstract and unfamiliar at once. However, I'd argue that such students are most likely already familiar with the abstract, through their own mathematical investigations. This is why often big universities have two mathematics major tracks, one geared towards prepping for graduate school and one for future high school math teachers and others who are interested in math but aren't ready for the really abstract stuff.
*Note I am taking liberty here and changing the question to "why isn't intro to proof writing emphasized in vector calculus as frequently as it used to be?" based on several people's comments on how "Advanced Calculus" is ill-defined. Proof based vector calculus seems to be the type of class Fantini is referring to as Advanced Calculus.
