Ah, that's a pity. It doesn't look like anyone else cites the lecture notes either...

Did you ever receive a response from Street?

While this is technically correct, I don't feel it is quite in the spirit of the question, since L-finite diagrams are the saturation of the finite diagrams, and so one doesn't obtain any new limits by considering them over the finite diagrams.

Yes, it's Proposition 2.5 of Relative monadicity.

One very simple observation is that the category of algebras admits limits when $D$ admits limits. So the problem reduces to finding conditions for which the category of algebras admits power objects.

When $\mathscr V = \mathrm{Set}$, admitting a $\mathscr V$-enriched presheaf category entails that the presheaf category is locally small (i.e. $\mathrm{Set}$-enriched), which is not the case for $[\mathrm{Set}^{\mathrm{op}}, \mathrm{Set}]$.

@მამუკაჯიბლაძე: sorry, I meant to write "symmetric/braided strict monoidal category", i.e. the braiding is not strict.

@მამუკაჯიბლაძე: these are the pseudoalgebras for the free strict symmetric/braided monoidal category 2-monads on $\mathbf{Cat}$.

My interpretation was that the first bullet point is intended to be read as "other limit-colimit-commutation classes containing the finite limits", and the second bullet point should include "where $\Phi$ contains the finite limit diagrams" (but this should be clarified in the question).