Indeed, the class of accessible categories is closed under lax limits, not under limits. I do understand that your homotopy pullback proves that the class of composables maps (f,g) with $f.g=\varnothing \to X$ is accessible. But the intersection with the image of the functor $T$ does not answer the question. How do you choose $T$ ? Where does it come from ?

There is a similar (but not sure that it is simpler) problem. In the category of $\Delta$-generated spaces, is the class of cofibrant contractible spaces accessible ? My intuition tells me "yes".

I don't understand your proof. Do you mean the map $\mathcal{C}^{[2]}\to \mathcal{C}^{[1]}$ which takes the composite ? And even with that, I still do not understand. By cofibrant replacement of $X$, I mean a pair $(X,f:Y\to X)$ where $Y$ is cofibrant and $f$ is a weak equivalence (I am also interested in the more restricted definition $f$ trivial fibration). And why use homotopy limits ? I believe (but I may be wrong) that the class of accessible categories, unlike locally presentable ones, is closed under a lot of operations like limits.

I added the reason in my question.

The paper you mention even gives the answer: top of page 24 : "We should remark that any locally presentable topologically bicomplete category also satisfies our hypothesis". And as far as I can understand the paper, any convenient category of topological spaces is fine, by convenient it is meant cartesian closed and containing enough topological spaces (e.g. CW-complexes).

Here is the beginning of an idea: by right-Bousfield localizing by all trivial fibrations, you will reduce the class of cofibrations. So the trivial fibrations must be interpreted as colocal equivalences. So I suggest first a right Bousfield localization by the set of simplices, and then if the new model category has exactly the monomorphisms as cofibrations (?), it should be "between" the minimal model structure and the usual model structure by Cisinski's result (so it should be left proper), then a left Bousfield localization by the accessible class of weak equivalences could work.

It is a shortcut for denoting the map $\mathcal{M}(Y,T)/\simeq \rightarrow \mathcal{M}(X,T)/\simeq$ if $f:X\rightarrow Y$.