Why are H+ and OH- ions preferentially discharged in the electrolysis of NaCl(aq)?
Water does not need to be dissociated into ions in order to participate in electrochemical reactions. While it is common to refer to electrode reactions as "discharging" this or that species, there is no requirement that the reactant of an electrode reaction must always be an ion.

In neutral solution, it's quite common for the water electrolysis reactions to be written as:

$$
\begin{align}
\text{Cathode:} &amp;\quad \ce{4H2O + 4e- -&gt; 2H2 + 2OH-} \\
\text{Anode:} &amp;\quad \ce{2H2O -&gt; O2 + 4H+ + 4e-}
\end{align}
$$

It's absolutely the case that the electrochemistry written using the ions as reactants in many cases is likely to be more facile:

$$
\begin{align}
\text{Cathode:} &amp;\quad \ce{4H+ + 4e- -&gt; 2H2} \\
\text{Anode:} &amp;\quad \ce{4OH- -&gt; O2 + 2H2O + 4e-}
\end{align} 
$$

But it's still far (far) more favorable for $\ce{H2O}$ to be reduced than $\ce{Na+}$, and in many cases more favorable for $\ce{H2O}$ to be oxidized than $\ce{Cl-}$. You're quite right that the concentrations of $\ce{H+}$ and $\ce{OH-}$ ions are extremely low in neutral salt solutions, but that fact is mostly irrelevant to the electrochemistry.