Old proof: Let's produce a series of values v_1, v_2, ..., such that v_n has at least n distinct prime factors. We can do this by noting that v_n + 1 has at least one prime factor but no prime factors in common with v_n, and thus taking v_{n + 1} to be v_n * (some prime factor of (v_n + 1)).

"New" proof: Let's produce a series of values v_1, v_2, ..., such that v_n has at least n distinct prime factors. We can do this by noting that v_n + 1 has at least one prime factor but no prime factors in common with v_n, and thus taking v_{n + 1} to be v_n * (v_n + 1).

Doesn't seem terribly different or more constructive or any such thing to me. I'd say both of these have the same fundamental content, just framed slightly differently.

[Just to head off what might superficially seem to be a significant distinction: Note that, in both proofs, we use the (constructive) fact that every integer > 1 has a prime factor, and even in the "new" proof, in order to actually extract an infinite stream of primes, one must actually carry out this process of finding prime factors for integers > 1 on demand.]