I think you're getting to the crux of the point. A few years in university (for undergrad) is not nearly enough time to cover the entirety of mathematics. The sheer fact of time constraints requires a survey and a selection of a fraction of the field for study.

I would argue that economics degrees are also, essentially, math degrees for the same reason. Neither economics degrees nor computer science degrees will cover the same topics as a "pure" mathematics degree (or else they would simply be known as mathematics degrees), and I will grant to you that they are "applied" math degrees if I must. But the point remains: a "pure" math undergrad degree cannot include all of mathematics, and thus is perhaps best renamed as theoretical mathematics, commensurate with branding economics and consumer science as "applied".