Adjoint functor theorem involves solution set condition and somehow sneaks in some category of elements or comma categories. I hate this way of doing things.
Is there any alternate way of doing it, say via Kan extension?
Adjoints can be expressed as Kan extensions,
$\operatorname{Ran}_{\mathcal{G}} I_{\mathcal{B}} \cong \mathcal{F}\dashv \mathcal{G}\cong \operatorname{Lan}_{\mathcal{F}} I_{\mathcal{A}}.$
I was thinking existence of adjoints can be translated into existence of some special Kan extensions, then for the coend formula to make sense, certain conditions will have to be met, this would turn out to be the solution set condition somehow?
The problem -in short, and quite roughly speaking- is that you need some assumptions ensuring that the colimit/coend to compute the left Kan extension $Lan_F1$ exists, as the domain might be a large category.
Conditions like the solution set condition or other forms of adjoint functor theorem are usually of the form "the colimit you would compute in order for the adjoint to exist can actually be computed on a small category, so it does exist".