For any n, there are only a finite number of hypergraphs on n vertices, much less linear hypergraphs. It is therefore correct, and equivalent to say that the proof omits "only a finite number of hypergraphs," when the result is shown for sufficiently large n. I haven't read the paper through, but, I suspect it's even possible to extract some explicit value of n for which the conjecture is true. In any case, it's enough to know that there is such an n.
I looked up Artin's primitive root conjecture on Wiki because I wasn't familiar with the result. I think you may be referencing the result that there are at most 2 primes for which the conjecture fails, no? If so, I would agree that's not close to a complete proof.
The difference here is that, in principle, since hypergraphs are just finite combinatorial objects, one could, in theory just run a generic coloring algorithm over the finite number of things left out of the proof to obtain the complete result with a bow tied around it.
In the case of Artin's conjecture, you have two difficulties: identifying 1 or 2 candidate primes for which it may fail, and verifying the conjecture for those primes. Even if you can somehow overcome the first difficulty and identify the candidate primes for which it may fail, in the absence of machinery that lets us make such a conclusion for those two primes, we are reduced to calculating an infinite number of discrete logarithms, which, obviously doesn't fly.
So, TL;DR: the difference here is literally the difference between finite and infinite. There are a finite number of hypergraphs left out, which can, in theory be tested using a finite number of operations to verify the conjecture. Not so with Artin's conjecture.
https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primit...