The hypergeometric functions arise naturally in the study of second order differential equations and, therefore, courses in mathematical physics. See "A Catalogue of Sturm-Liouville differential equations" by Everitt and "PDEs, ODEs, Analytic Continuation, Special Functions, Sturm–Liouville Problems and All That" by Burgess. Since they are related to second order equations, there is a connection to the Schwarzian derivative and all that entails. For a history of the HGFs, see Linear Differential Equations and Group Theory from Riemann to Poincare by Jeremy Gray.

The confluent hypergeometric functions include the families of Laguerre polynomials from which the Hermite polynomials can be constructed. These appear not only in quantum physics, probability theory, and analysis involving the heat, harmonic oscillator, and Schrodinger equations but also combinatorics. They also have associated ladder operators which form instances of a Graves/Lie/Heisenberg/Weyl algebra. Operators associated with the Laguerre polynomials can be related to a Witt-Lie algebra as well. OEIS A131758 has associations among the confluent hypergeometric functions and basic functions and number sequences arising in number theory and topology/characteristic classes. Special functions of the hypergeometric type are rife in group/representation theory and operational calculus, explored by Miller, Gilmore, Vilenkin, Carlitz, el-Salaam, Rota, Askey, Wilson, among others. Peruse also the book Hypergeometric Functions, My Love by Yoshida.

From the book A = B by Petkovsek, Wilf, and Zeilberger on hypergeometric identities important in general combinatorics and analysis of algorithms:

"Hypergeometric series are very important in mathematics. Many of the familiar functions of analysis are hypergeometric. These include the exponential, logarithmic, trigonometric, binomial, and Bessel functions, along with the classical orthogonal polynomial sequences of Legendre, Chebyshev, Laguerre, Hermite, etc.

It is important to recognize when a given series is hypergeometric, if it is, because the general theory of hypergeometric functions is very powerful, and we may gain a lot of insight into a function that concerns us by first recognizing that it is hypergeometric, then identifying precisely which hypergeometric function it is, and finally by using known results about such functions."

So, hypergeometric functions are studied by mathematicians and physicists but rarely in their full generality in classes—a symptom of specialization, motivated/rationalized more often than not by economics and an assembly-line, industrial mindset towards education.

(Dyson's dichotomy is needlessly polarizing. Dyson, who was proud to point out he had no Ph.D., unified Feynman's and Schwinger's approaches to QED and was subsequently awarded tenure at Princeton's IAS. He was perhaps the proverbial frog with wings. The greats like Riemann and John von Neumann break the mold, being shape-shifters morphing between the eagle and the jaguar, prowling both the skies and the hidden jungle beneath the canopy.)

Edit (4/28/20)

Addressing a question by Matt F and the continuing relevance of HGFs to modern mathematics, I'd like to point out relations among generalized differential operators and the confluent HGFs illustrated in the MO-Q&A "Pochhammer symbol of a differential and hypergeometric polynomial" and the paper "Contributions to the hypergeometric function theory of Heckman and Opdam: sharp estimates, Schwartz space, heat kernel" by Schapira (2018).

I first glimpsed the relation between the Pochhammer symbol/rising factorial and the diff op reps of the Kummer HGFs in the books on generalized functions by Gelfand and Shilov. Members of a family of KHGFs and families of KHGFs are related via conjugation of generalized diff ops, i.e., Rodriques-style formulas. For integer parameters, such diff op reps are related to important combinatoric constructs, such as the Dobinski formula as discussed by Rota in relation to the Stirling numbers of the second kind. See also, e.g., "Notes on the Rodrigues formulas for two kinds of the Chebyshev polynomials" by Feng Qi, Da-Wei Niu, Dongkyu Lim (2020). The importance of Laguerre, Hermite, Legendre, Chebyshev, etc. in diverse areas of math and physics ranging from analytic number theory to combinatorics to string theory and topology is well-documented.

I'm currently revisiting the lit on the connections between extremum principles and differential equations, particularly, the heat equation. The paper by Schapira cited above illustrates the continuing interest in these and associated topics, based on the HGFs presented in "Root systems and hypergeometric functions I" by Heckman and Opdam (1987).

15more comments