I would definitely be interested to know if there are any other such simple proofs exist for other theorems. (i.e. ones where a new proof simplifies it massively by using a simpler property).
The proof just given is conceptually even simpler than the original proof due to Euclid, since it does not use Eudoxus’s method of“reductio ad absurdum,” proof by contradiction. And unlike most other proofs of the theorem, it does not require Proposition 30 of Elements (sometimes called “Euclid’s Lemma”) that states: if p is a prime and p|ab, then either p|a or p|b. Moreover, our proof is constructive, and it gives integers with an arbitrary number of prime factors.
Edit: Actually, even though the article seems to imply that the classic proof ("most proofs") uses prop 30, it doesn't really seem to.
Euclid's proof of the infinitude of the primes was not phrased in terms of an overarching reductio ad absurdum (and even had it counterfactually been, mathematicians would've long ago been able to trivially rephrase it so as not to be, showing "For any finite set of primes, there is some further prime" directly).
And the classic proof of the infinitude of the primes does not anywhere use Proposition 30 of the Elements (see for yourself at http://aleph0.clarku.edu/~djoyce/elements/bookIX/propIX20.ht... ; Proposition 31 (that every composite has some prime factor) is used, but this in turn is argued for without any invocation of Proposition 30).
Where in the classic proof would you imagine "if p is a prime and p | ab, then p | a or p | b" would come up?