@SimonHenry: I realised I made a mistake after posting the comment. What I meant was that I suspect in this case the Eilenberg–Moore category is the $\mathbf{Ind}$-completion of the completion under coequalisers of T-algebra homomorphisms, of the subcategory of the Kleisli category of the monad on the finitely presentable objects.

It is certainly true that the Eilenberg–Moore category of a finitary (i.e. filtered-colimit preserving) monad on an accessible category (i.e. Ind-category) is itself accessible (i.e. an Ind-category).

Thank you, this is the observation I had overlooked. Could you add the elaboration in these comments to your answer? Then I'll accept it :)

It's still not clear to me why it suffices to consider only the faithful functors. I'm interested in locally fully faithful functors: why is it still sufficient if we drop fullness?

I'm probably missing something obvious, but why does it suffice to show that faithful functors are not closed under pushout? Why faithful and not fully faithful (which are closed under pushout)?

Nunes's Semantic Factorization and Descent may be related to what you are looking for (in particular, the "semantic factorisation" described within).

@ZhenLin: that's an interesting question. I'm not familiar enough with derived functors to know whether this would be a reasonable example. If there are examples of derived functors that people care about that are not absolute/pointwise, then it seems like a good example. But if in practice, one could define derived functors in terms of absolute Kan extensions without losing motivating examples, then probably not.

@TimCampion: I meant that we can construct Kan extensions specifically for the purpose of being examples of nonpointwise Kan extensions (one example being Exercise 3.9.7 of Borceux, though your examples probably also qualify in this sense), rather than being constructed for some useful purpose.

Categories without units are called semicategories. You can enrich a semicategory in a semigroupal category, which is what you describe. The Yoneda lemma is subtle with semicategories, but see On regular presheaves and regular semi-categories.

@JoMo: yes, sorry, I got the notation mixed up.

@DavidRoberts: Bunge suggested Pitts's 1985 "On product and change of base for toposes", but we have since found earlier references.

@SimonPepinLehalleur: thank you for this reference. I agree that it seems likely they knew about the presheaf case in light of this proposition, and it's certainly an earlier reference for some form of cocompletion based on presheaves.