Let $f$ be Riemann integrable on $[0,1]$ such that $\int_0^1 f(x)dx$ is nonzero. Show that there exists some interval such that $|f|>\alpha$ where $\alpha$ is a strictly positive real number.
Is the following correct?
Without loss of generality, assume the function $f$ is nonnegative. Then by Riemann integrability there exists a partition $P$ such that $L(f,P)>\int_0^1 fdx-1/2(\int_0^1 fdx)$. Therefore there exists some interval such that $|f|>1/2(\int_0^1 fdx)$ throughout the interval.
Just note that $|f| $ is also integrable with a positive integral and thus the lower Darboux sums for $|f|$ converge to a positive value. It follows that there is some sub-interval where $\inf |f|$ is positive. This is essentially same as your approach but we don't need to assume non negative $f$, that role is taken by $|f|$.