Why was the VC dimension not defined for all configurations of $d$ points? Let's start with a typical definition of the VC dimension (as described in this book)

Definition $3.10$ (VC-dimension) The $V C$ -dimension of a hypothesis set $\mathcal{H}$ is the size of the largest set that can be shattered by $\mathcal{H}$ :
$$
\operatorname{VCdim}(\mathcal{H})= \max \left\{m: \Pi_{\mathcal{H}}(m)=2^{m}\right\}
$$

So, if there exists some set of size $d$ that $\mathcal{H}$ can shatter and it cannot shatter any set of size $d+1$, then the $\operatorname{VCdim}(\mathcal{H}) = d$.
Now, my question is: why would we be just interested in the existence of some set of size $d$ and not all sets of size $d$?
For instance, if you consider one of the typical examples that are used to illustrate the concept of the VC dimension, i.e. $\mathcal{H}$ is the set of all rectangles, then we can show that $\operatorname{VCdim}(\mathcal{H}) = d = 4$, given that there's a configuration of $d=4$ points that, for all possible labellings of those points, there's a hypothesis in $\mathcal{H}$ that correctly classifies those points. However, we can also easily show that, if the 4 points are collinear, there's some labelling of them (i.e. the 1st and 3rd are of colour $A$, while the 2nd and 4th are of colour $B \neq A$) that a rectangle cannot classify correctly.
So, the class of all rectangles can shatter some sets of points, but not all, so we would need another class of hypotheses to classify all sets of four points correctly. The VC dimension does not seem to provide any intuition on which set of classes would do the trick.
So, do you know why the VC dimension wasn't defined for all configurations of $d$ points? Was this just a need of Vapnik and Chervonenkis for the work they were developing (VC theory), or could have they defined it differently? So, if you know the rationale behind this specific definition, feel free to provide an answer. References to relevant work by Vapnik and Chervonenkis are also appreciated.
I would say, that meaning of VC dimension is at least a possibility to implement any function on the given number of points for some case, not an express any function on $n$ points.
Yes, you are right, that this definition, unfortunately, is not very useful in practice.
Say, family of functions $\text{sign}(\sin(ax))$ has infinite $\text{VC}$ dimension. There exists an infinite sequence $x_n$, such that by tuning the parameter $a$ one can implement any logical function $0 1 0 \ldots 1$ on these points. However, this doesn't make this class of functions an outstanding ML algorithm.