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There are reason to doubt that the size of $\zeta(s)$ alone could be responsible for the location of the zeros. Let me present one argument in this direction: Suppose that the following (unlikely, but currently not ruled out) configurations of zeros occur in infinitely many intervals $[T; T + 1]$: we have roughly $\asymp \log T / \log\log T$ clusters of $\log\log T$ zeros, then in such interval $\zeta(s)$ should be of size $\exp(c \log T / \log\log T)$ (in particular contradicting the conjecture of Farmer, Gonek and Hughes). And then imagine that there are a few (say $4$) zeros of $\zeta(s)$ lying off the critical line. The two behaviors are envisage-able to occur simultaneously, unless of course we prove the falsehood/truth of each statement independently.