The following argument may not be the most direct one, but follows as a quick consequence of the characterization of Riemann integrability, which is perhaps the main result of the theory.
For a bounded function $f:[a,b]\to\mathbb{R}$ and for $\lambda >0$ let's denote
$$J _ \lambda :=\{ x\in [a,b]\, : \,\limsup _{y\to x}f(y)-\liminf _{y\to x}f(y) \ge \lambda\}\,.$$
The key point is that for any $\epsilon > 0$ there are two partitions $P^*$ and $P _ *$ of the interval $[a,b]$, both with mesh less than $\epsilon$, such that the corresponding left sums $s_L(f,P^*)$ and $s_L(f,P _ * )$ differ for more than $\lambda\operatorname{meas} (J _ \lambda)-\epsilon$ (to construct $P^*$ and $P _ *$, start from a covering of $J _ \lambda$ by a finite collection of disjoint intervals of length less than $\epsilon$, and whose left endpoints belongs to $J_\lambda$) . As a consequence
$$\limsup _ {|P|\to0} s_L(f,P) - \liminf _ {|P|\to0} s_L(f,P) \ge \lambda\operatorname{meas} (J _ \lambda)\, , $$
and we conclude that if $f$ is left integrable in the Cauchy sense, then $\operatorname{meas} (J _ \lambda)=0$ for all $\lambda >0$, that is, it is continuous a.e., hence Riemann integrable by the characterization.