My teachers never accepted this demonstration. Their point was (from wikipedia:

Euclid is often erroneously reported to have proved this result by contradiction, beginning with the assumption that the finite set initially considered contains all prime numbers, or that it contains precisely the n smallest primes, rather than any arbitrary finite set of primes.[4] Although the proof as a whole is not by contradiction (it does not assume that only finitely many primes exist), a proof by contradiction is within it, which is that none of the initially considered primes can divide the number q above.

I always thought this was complicated for no special reason so I prefer the one you show. I find more natural.