How does "3 is prime" imply that "3" exists, while "primeness is membership in the set of zeroes of the Zeta function" not imply that "primeness" exists?

The latter does imply primeness exists. But "3 is prime" doesn't. Luckily you haven't just used primeness as the value of a bound variable, you've given an appropriate paraphrase (although now you're committed to the existence of the set of zeros of the Zeta function).

This whole discussion is, ultimately, about the definition of the word "exist". But if you try to hang that definition off linguistic phenomena, then you're at the whim of every linguistic construct people can come up with, and people can probably twist words to get something covered by your definition that you didn't want to be.

Huh?