What are applications of computational models beyond P?
Not all problems that are encountered in an industrial setting are clearly either NP-Complete or else polynomial-time solvable, even with randomness, heuristics and approximation. So if you encounter (a variant of) a problem such as Exact Independent Set, or Succinct Graph Diameter, for example, then you ought to be looking to the complexity classes $\text{DP}$ and $\Sigma_2^P$. Without the study of these classes, there is no proper placement of these problems, no way to relate them to one another and therefore no way to tackle them coherently, and no way to quantify their hardness. We might look for a way to compactly store and then sell the solutions to NP-Complete problems after we've brute forced them for many inputs. But because of Karp and Lipton, we know that this is impossible, unless the polynomial hierarchy collapses to the second level. In general, these conditional collapses are great for question of the form "Can I...", "No.".

I would love to provide an answer like "We were struggling, but then Immerman and Szelepcsényi proved that $\text{Nondeterministic Logspace}$ was closed under complement, and that really gave a boost to our startup!", being a student myself I don't have any anecdotal evidence, and I would love to hear from someone more seasoned.

So if you are looking for an industrial application of computational complexity theory beyond SAT Solvers, then you may be disappointed. Complexity theory is to computer science a bit like string theory is to physics; it is purely theoretical and its study is not motivated by the urge to solve a particular problem, but rather to understand the nature of computation - how do randomness and nonuniform computation relate to one another? Do all programs with small memory footprints have a fast implementation? - very much like the motivation of theoretical physics is not to design a slightly more efficient engine, but rather to understand the laws of physics. You are taking a theoretical class, and must not be surprised to learn about theoretical results. There are coding classes where you come from; if you prefer those, take those.

If I take the liberty of venturing into speculation, then you may know that right now we do not even know how to rule out ridiculous scenarios in which $\text{BPP}=\text{NEXP}$. Of course everyone believes that the result will be $\text{BPP}\subsetneq\text{NEXP}$, but a proof of this fact cannot come without major advances in our understanding of computation, and, staying with the metaphor of physics, where major advances in our understanding of fundamental physics invariably spurs major progress in unexpected areas, I feel confident in saying that these results, when they come, will spur major progress across all fields of computing.