Apart from the applied stuff you mention, the real core of a mathematics education involves, I think, 4 main areas with significant overlap
Group A: number theory, graph theory, combinatorics
which shares concepts with
Group B: Algebra, Topology, complex analysis, differential geometry, metric spaces...etc
which shares concepts with
Group C: Functional analysis, measure theory
which shares concepts with
Group D: probability and statistics.
As for the applied math that you mention, you should really need to add vector calculus and I'd highly encourage anyone to take a course on fluid mechanics (from a mathematics department instead of an engineering department) to get a real feel for vector calculus in action.
Real analysis, complex analysis, topology, and number theory are there (topology and number theory are both listed as electives since most math programs categorize them as such). Graph theory, functional analysis, differential geometry, probability, and statistics are almost always either electives or graduate courses.
It’s funny, because most of the things you mention as “real math” are things that many math undergraduates don’t learn (not until graduate school at least) but that physics students learn as undergraduates (differential geometry, measure theory, functional analysis, etc.).
though I am a little surprised that they have 1 course of differential equations in there instead of complex analysis as a required topic, as I think the latter is a better pure math topic. But it's MIT, so be it. Whether directly or indirectly, many of us learned to view math the MIT way by patiently working through foundational books like Artin and Munkres.
That said, my mention of non-introductory algebra topics probably is more of a personal idiosyncracy/interest.
