I think you're essentially correct that analytic geometry is not considered a worthy topic of study for a contemporary US math major, or at least not worthy enough to be part of the standard curriculum.
I hear that things used to be quite different: in the first half of the 20th century math majors spent much of their career learning analytic geometry and only learned calculus in their junior or senior year. (I learned this from Paul Halmos's autobiography, which I highly recommend to all.)
Nowadays someone my age (I am in my early to mid thirties) or younger need not know what "analytic geometry" means at all: it is that much of a forgotten field. I feel like I got a sort of brief window into the past by virtue of a self-paced math course I took in the CTY program (as a high school student). In such courses you work through an entire textbook by yourself. It turns out that then when you actually read a high school math textbook from cover to cover you learn lots of things which are not covered in most actual high school courses: perhaps most notably, I assumed ever after that mathematical induction was part of the algebra two curriculum because it was in the textbook I read for that subject. (Well, not ever after. In the last ten years of teaching freshman calculus, I've found that maybe one student in 200 has encountered induction in their high school curriculum.) Eventually I got to "analytic geometry", which was sort of ten times more about conic sections than I really wanted to know: foci, directrices, something called the latus rectum, and so forth. I wasn't that thrilled with it, to be honest. I especially remember that sometimes you got conics with "cross terms" and then you had to rotate axes in order to deal with them. This topic I did revisit later on in a linear algebra course, and I agree that it's kind of ridiculous to do it without linear algebra: it's both much harder and much less clear what's going on.
I think most of the "analytic geometry" that we do see nowadays is in the geometric applications that one often does a little bit of in either linear algebra or multivariable calculus. In fact, exactly why linear algebra is entirely sufficient for analytic geometry is not clear to me, since linear algebra is most directly concerned with linear subspaces and in analytic geometry one studies certain plane and space curves. I can see that a connection is provided by the theory of quadratic forms, but -- depending once again on the scope of what you mean by analytic geometry (I have never seen an intrinsic definition of the subject; to me it's just a collection of topics) -- I would want to use some non-linear algebra (e.g. basic projective geometry like Bezout's theorem) and multivariable calculus.