The question is a bit imprecise, so I'm not quite sure whether this will really answer it, but let me try to paint the reduction of weighted colimits to conical colimits in a more "conceptual" light.

First of all, your identity $[\mathcal{A},\mathsf{Set}](\Delta \ast, \mathcal{B}(X,F)) \cong [\mathcal{A},\mathcal{B}](\Delta X, F)$ is correct; we can write it out in terms of ends:

$\int_{A \in \mathcal{A}} [\Delta \ast(A), \mathcal{B}(X,F(A))] = \int_{A \in \mathcal{A}} \mathcal{B}(X,F(A)) = \int_{A \in \mathcal{A}} \mathcal{B}(\Delta X(A), F(A))$

Now, let me argue that the key identity is $G = \operatorname{Lan}_{\phi_G} (\Delta \ast)$. This identity implies that the $G$-weighted limit of $F$ is the $\Delta \ast$-weighted limit of $F \phi_G$, because we have

$[\mathcal{A},\mathsf{Set}](G, \mathcal{B}(X,F)) = [\mathcal{A},\mathsf{Set}](\operatorname{Lan}_{\phi_G} (\Delta \ast), \mathcal{B}(X,F)) = [\mathcal{A},\mathsf{Set}](\Delta \ast, \mathcal{B}(X,F\phi_G))$

That is, the defining property of a left Kan extension relates the functor represented by a weighted limit, on the left, to the functor represented by the conical limit, on the right.

So from this perspective, the key property of the category of elements is that it allows us to express our copresheaf as the left Kan extension of a constant-at-1 presheaf. I'm not sure whether this is a universal property, but it is reminiscent of the universal property of a subobject classifier, and of a universal property the category of elements does possess -- it is the classifying discrete opfibration. We have a diagram:

$\require{AMScd}$ \begin{CD} \mathsf{el}G @>>> \ast \\ @V \phi_G V V @VV \ast V\\ \mathcal{A} @>>G> \mathsf{Set} \end{CD}

The property I'm pointing out is that the bottom is a left Kan extension, while the usual universal property says that this is a pullback comma square. I'm not quite sure of the relationship.

The question is a bit imprecise, so I'm not quite sure whether this will really answer it, but let me try to paint the reduction of weighted colimits to conical colimits in a more "conceptual" light.

First of all, your identity $[\mathcal{A},\mathsf{Set}](\Delta \ast, \mathcal{B}(X,F)) \cong [\mathcal{A},\mathcal{B}](\Delta X, F)$ is correct; we can write it out in terms of ends:

$\int_{A \in \mathcal{A}} [\Delta \ast(A), \mathcal{B}(X,F(A))] = \int_{A \in \mathcal{A}} \mathcal{B}(X,F(A)) = \int_{A \in \mathcal{A}} \mathcal{B}(\Delta X(A), F(A))$

Now, let me argue that the key identity is $G = \operatorname{Lan}_{\phi_G} (\Delta \ast)$. This identity implies that the $G$-weighted limit of $F$ is the $\Delta \ast$-weighted limit of $F \phi_G$, because we have

$[\mathcal{A},\mathsf{Set}](G, \mathcal{B}(X,F)) = [\mathcal{A},\mathsf{Set}](\operatorname{Lan}_{\phi_G} (\Delta \ast), \mathcal{B}(X,F)) = [\mathcal{A},\mathsf{Set}](\Delta \ast, \mathcal{B}(X,F\phi_G))$

That is, the defining property of a left Kan extension relates the functor represented by a weighted limit, on the left, to the functor represented by the conical limit, on the right.

So from this perspective, the key property of the category of elements is that it allows us to express our copresheaf as the left Kan extension of a constant-at-1 presheaf. I'm not sure whether this is a universal property, but it is reminiscent of the universal property of a subobject classifier, and of a universal property the category of elements does possess -- it is the classifying discrete opfibration. We have a diagram:

$\require{AMScd}$ \begin{CD} \mathsf{el}G @>>> \ast \\ @V \phi_G V V @VV \ast V\\ \mathcal{A} @>>G> \mathsf{Set} \end{CD}

The property I'm pointing out is that the bottom is a left Kan extension, while the usual universal property says that this is a pullback comma square. I'm not quite sure of the relationship.

EDIT Just an additional miscellaneous thought: if this were also true in the enriched setting, then this would tell us for example that every group representation was the induced representation of a trivial representation of some algebra (or rather, algebroid -- and of course, "trivial representation" doesn't even make sense in this generality). This isn't true, but it's still a good strategy to look for representations as subrepresentations of induced representations of trivial representations.

The question is a bit imprecise, so I'm not quite sure whether this will really answer it, but let me try to paint the reduction of weighted colimits to conical colimits in a more "conceptual" light.

First of all, your identity $[\mathcal{A},\mathsf{Set}](\Delta \ast, \mathcal{B}(X,F)) \cong [\mathcal{A},\mathcal{B}](\Delta X, F)$ is correct; we can write it out in terms of ends:

$\int_{A \in \mathcal{A}} [\Delta \ast(A), \mathcal{B}(X,F(A))] = \int_{A \in \mathcal{A}} \mathcal{B}(X,F(A)) = \int_{A \in \mathcal{A}} \mathcal{B}(\Delta X(A), F(A))$

Now, let me argue that the key identity is $G = \operatorname{Lan}_{\phi_G} (\Delta \ast)$. This identity implies that the $G$-weighted limit of $F$ is the $\Delta \ast$-weighted limit of $F \phi_G$, because we have

$[\mathcal{A},\mathsf{Set}](G, \mathcal{B}(X,F)) = [\mathcal{A},\mathsf{Set}](\operatorname{Lan}_{\phi_G} (\Delta \ast), \mathcal{B}(X,F)) = [\mathcal{A},\mathsf{Set}](\Delta \ast, \mathcal{B}(X,F\phi_G))$

That is, the defining property of a left Kan extension relates the functor represented by a weighted limit, on the left, to the functor represented by the conical limit, on the right.

So from this perspective, the key property of the category of elements is that it allows us to express our copresheaf as the left Kan extension of a constant-at-1 presheaf. I'm not sure whether this is a universal property, but it is reminiscent of the universal property of a subobject classifier, and of a universal property the category of elements does possess -- it is the classifying discrete opfibration. We have a diagram:

$\require{AMScd}$ \begin{CD} \mathsf{el}G @>>> \ast \\ @V \phi_G V V @VV \ast V\\ \mathcal{A} @>>G> \mathsf{Set} \end{CD}

The property I'm pointing out is that the bottom is a left Kan extension, while the usual universal property says that this is a pullback. I'm not quite sure of the relationship.

The question is a bit imprecise, so I'm not quite sure whether this will really answer it, but let me try to paint the reduction of weighted colimits to conical colimits in a more "conceptual" light.

First of all, your identity $[\mathcal{A},\mathsf{Set}](\Delta \ast, \mathcal{B}(X,F)) \cong [\mathcal{A},\mathcal{B}](\Delta X, F)$ is correct; we can write it out in terms of ends:

$\int_{A \in \mathcal{A}} [\Delta \ast(A), \mathcal{B}(X,F(A))] = \int_{A \in \mathcal{A}} \mathcal{B}(X,F(A)) = \int_{A \in \mathcal{A}} \mathcal{B}(\Delta X(A), F(A))$

Now, let me argue that the key identity is $G = \operatorname{Lan}_{\phi_G} (\Delta \ast)$. This identity implies that the $G$-weighted limit of $F$ is the $\Delta \ast$-weighted limit of $F \phi_G$, because we have

$[\mathcal{A},\mathsf{Set}](G, \mathcal{B}(X,F)) = [\mathcal{A},\mathsf{Set}](\operatorname{Lan}_{\phi_G} (\Delta \ast), \mathcal{B}(X,F)) = [\mathcal{A},\mathsf{Set}](\Delta \ast, \mathcal{B}(X,F\phi_G))$

That is, the defining property of a left Kan extension relates the functor represented by a weighted limit, on the left, to the functor represented by the conical limit, on the right.

So from this perspective, the key property of the category of elements is that it allows us to express our copresheaf as the left Kan extension of a constant-at-1 presheaf. I'm not sure whether this is a universal property, but it is reminiscent of the universal property of a subobject classifier, and of a universal property the category of elements does possess -- it is the classifying discrete opfibration. We have a diagram:

$\require{AMScd}$ \begin{CD} \mathsf{el}G @>>> \ast \\ @V \phi_G V V @VV \ast V\\ \mathcal{A} @>>G> \mathsf{Set} \end{CD}

The property I'm pointing out is that the bottom is a left Kan extension, while the usual universal property says that this is a pullback comma square. I'm not quite sure of the relationship.