Can I compute closest split pair of points where distance is strictly less than delta
Yes, as you have observed, we can just compute pair of points whose distances are strictly less than $\delta$ in the "combine" step of the "divide-conquer-combine" algorithm, where we examine the minimal distance between the points to the left and the points to the right. 

However, it is not enough to just compute the next 3 points.





The above graph, taken from CLRS, illustrates the original argument for 7 in the case of non-pairwise distinct points and for 5 in the case of pairwise distinct points. For the sake of clearer explanation, assume the central line $l$ is the vertical axis and $\delta=\delta_L=\delta_R=1000$. Let the strip consist of the area between the line $x=1000$ and $x=1000$, excluding those two lines. This strip contains only the points that are strictly less than 1000 away from the central line $l$. Suppose all distinct points in the strip in the ascending order of vertical coordinate are $s_0, s_1, s_2, s_3, s_4$.

Suppose we have verified that $d(s_0,s_i)\ge1000$ for $i=1,2,3$. Can we conclude that there is no point $s$ such that $d(s_0,s)&lt;1000$?

No. It might happen that
$$\begin{align}\\
s_0&amp;=(79,0)\\
s_1&amp;=(-921,1)\\ 
s_2&amp;=(999,500)\\  
s_3&amp;=(-999, 998)\\  
s_4&amp;=(79, 999)
\end{align}$$

Note that $d(s_0, s_4)= 999&lt;1000$. In fact, $d(s_i,s_j)\gt1000$ for all $i$ and $j$ except $d(s_0, s_4)$.
$$\begin{align}
d(s_0, s_1)&amp;=\sqrt{1000001}  \\
d(s_0, s_2)&amp;=\sqrt{1096400}  \\
d(s_0, s_3)&amp;=\sqrt{2158088}  \\
d(s_0, s_4)&amp;=\sqrt{998001}  \\
d(s_1, s_2)&amp;=\sqrt{3935401}  \\
d(s_1, s_3)&amp;=\sqrt{1000093}  \\
d(s_1, s_4)&amp;=\sqrt{1996004}  \\
d(s_2, s_3)&amp;=\sqrt{4240008}  \\
d(s_2, s_4)&amp;=\sqrt{1095401}  \\
d(s_3, s_4)&amp;=\sqrt{1162085}  \\
\end{align}$$
So if we only compute the distance with the next 3 points, we will not find that shortest edge from $s_0$ to $s_4$ at the end of the algorithm.

The above counterexample shows that it is not sufficient to just compare 3 following points of every point in a middle strip only containing points strictly less than $\delta$ from the center line, even if we assume that all points are distinct.