You made an error: p1p2...pn + 1 is not necessarily prime, but none of its prime factors are in {p1, ..., pn}. For example, 15 = 2 * 7 + 1 is not prime, but neither 3 nor 5 are in {2, 7}. I agree with your overall point though.
I said it was nitpicking because it is only remotely relevant to the article, but the distinction does matter in mathematics. See for instance this post[0], where this very proof is discussed in the comments.
Euclid's proof: Every finite set of primes {p1,p2,...pn} has a prime, namely p1p2...pn + 1, not in the set.
Contradiction proof: The hypothetical finite set of all primes {p1,p2,...pn} would have a prime, namely p1p2...pn + 1, not in the set.
I'd hardly call the latter a different proof.