Question regarding Cook-Levin theorem proof
No, it would not. The reason is that the encoding of configurations might change in 3 consecutive places with a single transition. Specifically, this happens with transitions where the head moves to the left.

For example, consider the configuration $aabqcaa$, which means that the tape contains $aabcaa$ and the head is on $c$. Now, assume that we have the transition $\delta(q,c)=\{(s,d,L)\}$. So the $c$ should be replaced with $d$, the head should move left, and the state should change to $s$. Thus, the new configuration is $aasbdaa$. You can see that the change is in 3 consecutive places.

Had you only tracked changes is 2x2 windows, you should have allowed (for example) $qc$ to turn to $bd$, meaning that you allow the head to just "disappear".