I don't think that's quite right. There are two possible precise statements for such claims:

1) There exists some n such that all integers >= n satisfy the desired property.

2) For n = [a specific constant], all integers >= n satisfy the desired property.

I'm not familiar with Artin's conjecture, but from your description, it satisfies (1) but not (2). The reason is that if there are at most 2 such primes, we can take n to be the larger of the two primes plus one. Since all primes are finite, this choice of n is also finite.

I think the key question is whether the paper described in this article proves (1) or (2). From the discussion, it sounds like it proves (2), which is the stronger result.