Why are random numbers important to the study and application of computer science?
There are a lot of applications of Random numbers in Computer science, there are full books about it, check Randomized Algorithms by Rajeev Motwani and Prabhakar Raghavan, for instance. There you will find many arguments for the case of random numbers.

Here is a not exhaustive list of applications in sub-fields of CS:


Algorithms (As Yuval pointed, there are many problems where only randomized algorithms are known and you will find that many times randomized algorithms are simpler and more efficient than their deterministic counterpart)
Machine Learning ( Generative algorithms, variational autoencoders, GANS, Bayesian inference, Monte-Carlo algorithms, ...)
Cryptography(randomized prime number testing, ...)
Computer graphics and simulation (Path Tracing, Procedural stuff, games, ...)
Big Data (search for sketching algorithms, probabilistic counting, Bloom filters, ...)
Optimization (stochastic gradient descend, meta-heuristics)
Simulation (random processes, multi-agent, stochastic differential equations,...)


I am sure there are plenty more.