Thank you, this is helpful! Is there reason to believe there may not be a nice characterisation for (1)?

@ZhenLin: just to clarify, what would be the corrected statement?

(AFT) is essentially present in Theorem 13 of The adjoint functor theorem and the Yoneda embedding (Ulmer, 1971), though one must unwind "every cocontinuous functor" to get the result in the case of a specific cocontinuous functor.

"I think there may not be much more to say at this level of generality" — it's also not clear to me. However, it doesn't seem unreasonable that there might be at least sufficient syntactic conditions that cover examples like the powerset monad and CHS monad. It is perhaps helpful to rephrase the condition that $T$ be varietal in these ways, but these reformulations are very direct, and it's not clear that there aren't stronger results at this level of generality.

@TimCampion: yes, that's right. Compact Hausdorff spaces is another example I would be interested in: I'll update the question, thanks.

Note that if a category with a terminal object is not cartesian-closed, it is also not locally cartesian-closed.

Yes, that's right.

Does this answer your question? $\infty$-ary tensor product on a category

@ZhenLin: whoops, thanks!