To go from the definition of the Riemann integral ($f$ is Riemann integrable on $[a,b]$ if there exists a real $A$ such that $\forall \epsilon >0, \exists \delta>0$ such that $\forall D=\{([a_i,a_{i+1}],x_i)\}$ tagged partition of $[a,b]$ $h_i=a_{i+1}-a_i < \delta \implies |S_D(f) - A| < \epsilon$, where $S_D(f)$ is the Riemann sum on $D$) to the definition of the KH integral, all we do is replace the constant $\delta$ by a ("variable") gauge function $\delta(x_i)$.
However, the definition of the Riemann integral allows arbitrary tagged partitions, as long as $h_i<\delta$. Isn't that as flexible as the definition of the KH integral ($h_i < \delta(x_i)$)? For a given $\epsilon$, we can first find a tagged partition that will make the Riemann sums converge, then take $\delta = max({h_i})$ and satisfy the Riemann definition? In other words, any tagged partition that is allowed in the definition of the KH integral also seems to be allowed in the definition of the Riemann integral, and making the $\delta$ variable doesn't seem to be "essential". Of course, I am missing something. What am I missing?