Are Primes a Self-Fulfilling Prophecy?

Leaving some primes out of the Euler product won't affect the location of the zeroes, since you will end up with the Zeta function multiplied by a non-zero analytic function (which won't produce any more zeros), and the formula for the prime counting function depends only on the location of the zeroes. So as far as I understand your algorithm, yes, it will "regenerate" any (finite number of) primes that were initially missing.

EDIT: Corrected my erroneous description of the missing term as "constant".

EDIT: To show that the analytic continuation of the product is the same as the product of the analytic continuation, use the fact that the analytic continuation is unique: "Let $f_1$ and $f_2$ be analytic functions on domains $\Omega_1$ and $\Omega_2$, respectively, and suppose that the intersection $\Omega_1 \cap \Omega_2$ is not empty and that $f_1 = f_2$ on $\Omega_1 \cap \Omega_2$. Then $f_2$ is called an analytic continuation of $f_1$ to $\Omega_2$, and vice versa (Flanigan 1983, p. 234). Moreover, if it exists, the analytic continuation of $f_1$ to $\Omega_1$ is unique." We will also need the fact that $\zeta(s)$ and $1-p^{-s}$ are analytic (and the more basic fact that the product of two analytic functions is analytic).

COMMENT: I think your idea here is pretty interesting! I suspect that it may work even in some cases where an infinite number of primes are discarded.