I was recently compiling some notes for an undergrad-level course on number theory, and I went over the proof of the fact that $(\mathbb{Z}/p\mathbb{Z})^\times$ is cyclic for any prime $p$: it's a finite abelian group and thus the direct sum of cyclic groups, and the fact that $X^r - 1\in (\mathbb{Z}/p\mathbb{Z})[X]$ has at most $r$ zeros forces $(\mathbb{Z}/p\mathbb{Z})^\times$ to be cyclic itself. It's a completely nonconstructive proof, and there aren't many tools available at that level for digging any deeper into the problem.

So, what is the current state of the art for primitive roots mod $p$? I'm not a number theorist, and I don't know much about the problem aside from Artin's conjecture (which, as far as I know, is still a conjecture). I think it's known that it holds for infinitely many primes unconditionally and all but a small, finite number modulo a particular form of the generalized Riemann hypothesis, but those results are both several decades old. (Even the most relevant questions on this site I could find are more than a decade old.) Are there any newer results or any new methods that look promising, or is this just an untractable problem for now? Or, for that matter, is the last result I mentioned considered a satisfying answer? (To clarify, I'm asking for my own benefit and am not looking for an undergrad-level answer.)

A Computational Introduction to Number Theory and Algebra: "there is no known way to efficiently recognize a primitive root modulo $p$ without knowing the prime factorization of $p − 1$." That is a statement being made over all primes $p$. $\endgroup$