The zeta function can be written $\zeta(1/2+it)=Z(t)e^{-i\vartheta(t)}$ where $Z(t)$ are real analytic and $\vartheta(t)$ is a simple function related to $\Gamma(s)$. The relation between $\zeta(1/2+it)$ and $\zeta'(1/2+it)$ can be written in a simpler form $$-2\vartheta'(t)\zeta(1/2+it)=\zeta'(1/2+it)+e^{-2i\vartheta(t)}\zeta'(1/2-it)$$ in which you may change $\zeta'(1/2-it)=|\zeta'(1/2+it)|^2/\zeta'(1/2+it)$. This is proved in page 222 of the paper ``On the exact location of the non-trivial zeros of Riemann's zeta function'', Acta Arithm. 163 (2004) 215--245.

The proof do not depend on the nature of $Z(t)$. So this ``differential equation'' is satisfied by $f(1/2+it) =e^{-i\vartheta(t)} u(t)$ with $u$ any even real analytic function. Therefore it has no information about the zeros of zeta.

The zeta function can be written $\zeta(1/2+it)=Z(t)e^{-i\vartheta(t)}$ where $Z(t)$ are real analytic and $\vartheta(t)$ is a simple function related to $\Gamma(s)$. The relation between $\zeta(1/2+it)$ and $\zeta'(1/2+it)$ can be written in a simpler form $$-2\vartheta'(t)\zeta(1/2+it)=\zeta'(1/2+it)+e^{-2i\vartheta(t)}\zeta'(1/2-it)$$ in which you may change $\zeta'(1/2-it)=|\zeta'(1/2+it)|^2/\zeta'(1/2+it)$. This is proved in page 222 of the paper ``On the exact location of the non-trivial zeros of Riemann's zeta function'', Acta Arithm. 163 (2004) 215--245.

The proof do not depend on the nature of $Z(t)$. So this ``differential equation'' is satisfied by $f(1/2+it) =e^{-i\vartheta(t)} u(t)$ with $u$ any even real analytic function. Therefore it has no information about the zeros of zeta.