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@TimCampion: I would also be interested to know if there are any such examples, if there are no nonposetal examples :) Maybe I'll cheekily update my question.
Thank you, this is already very strong evidence that atomocity of every object is an extremely restrictive assumption. In other words, every tiny object must be "superinitial" (assuming enough coproducts).
This is a beautiful idea, and I very much look forward to reading the paper when it is ready. Is there any relation to the fact that (unenriched) limits can be constructed using large colimits (e.g. Proposition 12.8 of The Joy of Cats)?
You may also be interested in this recent talk, which presents a novel perspective on the structure–semantics adjunction which has similar inspiration to your questions.
Unfortunately I still don't have time to expand, but Auderset's Adjonctions et monades au niveau des 2-catégories gives a fairly explicit derivation of the Eilenberg–Moore and Kleisli constructions from the 2-categorical perspective. Pumplün's is a different (albeit related) construction, but more relevant to the structure–semantics adjunction, because it also characterises the functors between Eilenberg–Moore categories and between Kleisli categories appropriately, which Auderset's does not.
A duality relative to a limit doctrine gives a general answer to "To what extent can Gabriel–Ulmer duality be generalised to different classes of limits and colimits?". It's not in the terminology of sketches, but that should be a straightforward conversion. The enriched version will follow from Accessibility and presentability in 2-categories.
@VladimirSotirov: that's an interesting observation, thanks! (Presumably this is something you spotted and isn't written down in the literature anywhere?)