...over my career, both as an academic and as professional software engineer, I appreciate fully the distinction made between the mathematical foundations of computer science and the application of computer solutions to mathematically oriented problems. To start a four-year degree in computer science with the same calculus-oriented math series that the "physical world" majors take is a bit wrong-headed IMHO, but not completely. First, the math of computing is discrete, and this deserves first attention in a good discrete math course right after college algebra. And for most of my career, a solid foundation in logic, sets, relations, etc. served me well in both professional software development and college teaching. Indeed, my schooling went as follows: BS CIS, MS CS, and DCS (that's Doctor of Computer Science, as opposed to PhD...), where my bachelor's program had both a solid business core as well as just enough "continuous math" to understand the foundations of calculus. Missing was the discrete math I mentioned above, but I got that in my MS.
But now, I find myself smack in the middle of the defense/aerospace business, and the day-job application involves aspects of both calculus and statistics for which my schooling did not fully prepare me. Now, my role is more about technical leadership than practition-ing, so I'm not floundering, but I've had to dig out the old texts and learn some math on my own that most of you learned (or slept through) in your earliest years of college, or even in high school. What's really important for me to understand are things like the computational complexity of a proposed solution, that a branching structure in a code segment covers both nominal and corner cases (they do let me sit in on peer reviews...), and other foundational computer science things that the schools, in their increasing "IT" orientation, aren't covering much anymore.
I was an academic advisor for a year, probably the worst on the planet, because I told my students things like, "major in CIS, then switch to CS for your masters, avoid the calc hell" and "don't be doing school unless you're really motivated in the major" (ha, the admissions advisors LOVED that one... NOT!)
So, if I were king, I'd make all computer science students take discrete math, so there. After that, the math depends on what industry (domain, applications, whatever) in which you plan to work. For some, that may be statistics, for others the calc series. But the point is that the primary math of all computer professionals is logic, sets, relations, and the rest of the "discrete phylum", and that should be learned for competency, first. Doggonit.