The Riemann hypothesis has many strong consequences in number theory. The question is: would a bound on the number of zeros of Riemann zeta-function in the critical strip with real part not equal 1/2 have significant consequences? For example, what would the statement that a number of such zeros is finite imply?



Consequences of a bound on possible counterexamples to Riemann hypothesis
Such bounds are generally called zero-density estimates (for the Riemann zeta function or more general $L$-functions), and they have significant consequences. Chapter 10 of Iwaniec-Kowalski's Analytic number theory is devoted to this topic. 


A famous example of such a result and application is Huxley's theorem (Inventiones Mathematicae 15 (1972), 164-170): for $x^{7/12+\epsilon}<h<x$ the number of primes in $[x,x+h]$ is asymptotically $h/\log x$. Other fascinating consequences include connections with the Lindelöf Hypothesis (see e.g. the work of Paul Turán).


The statement you propose is stronger than the famous [Density Hypothesis](https://www.encyclopediaofmath.org/index.php/Density_hypothesis) (which is also implied by the Lindelöf Hypothesis) for the Riemann zeta function. This hypothesis would imply the above statement with $1/2$ in place of $7/12$.


