In the paper "On the Level Curves of the Xi Function" http://arxiv.org/abs/1002.0352v8, John Breslaw takes a very similar approach to a study of the Riemann hypothesis I did while I was in my first year of university. And I still have a lot of literature stored away somewhere concerning this approach, I got bored and stopped working on the problem in this way. I'm now worried that perhaps I should of kept at it and published something similar in the early 2010.
Can someone please explain why this is not a legitimate way to approach the problem?
For all these approaches, you can check them against zeta-like functions that DO violate their Riemann hypothesis. If your reasoning doesn't distinguish these functions, obviously you can't distinguish between RH-satisfying and RH-violating functions. An easy example is the Davenport-Heilbronn function, $$ \mathcal{L}_5 = \frac{1}{2} \sec{\theta} \left( e^{-i\theta} L(s,\chi_1 ) + e^{i\theta} L(s,\chi_3) \right), $$ where $\chi_1$ and $\chi_3$ are two of the characters modulo $5$, and $\theta$ satisfies $$ \tan{\theta} = \frac{\sqrt{10-2\sqrt{5}}-2}{\sqrt{5}-1}.$$ You can find more information about this one in Titchmarsh's book on the zeta-function.
An even simpler, more recent example is due to Balanzario (Eugenio P. Balanzario. “Remark on Dirichlet series satisfying functional equations.” eng. In: Divulgaciones Matemáticas 8.2 (2000), pp. 169–175.), where you can consider $$ f_1(s) = (1+5^{1/2-s})\zeta(s), \qquad f_2(s) = L(s,\chi_2), $$ where again $\chi_2$ is a particular character modulo $5$. These satisfy the same functional equation, $$ f_i(s) = 5^{1/2-s} 2(2\pi)^{s-1} \Gamma(1-s) \sin{\tfrac{1}{2}\pi s} \, f_i(1-s), $$ but obviously we can look at $$ f_1(s)f_2(a)-f_2(s)f_1(a), $$ which certainly has a zero at $s=a$.