So, I think, a very simple, and 'essentially' unique, way to obtain such situation is to get model of $(\infty,n)$-category which are categories of either presheaf of set or presheaf of spaces over a small category $C$ of 'diagrams'. (You can generally jump from a description as presheaf of sets to a description as presheaf of spaces, using some variant of the simplicial completion techniques as illustrated for example in the case of the category $\Theta$ Here.)

While producing a rigorous argument for the converse will be difficult, it definitely feels also true: as soon as you have a rich enough class of diagrams $C$ with the sort of properties you are requiring one should be able to prove some sort of Nerve theorem (an $\infty$-categorical version of monads with arities or Nervous monads) too shows that $(\infty,n)$-categories can be represented as presehaf of spaces on $C$ satisfying some segal conditions, for which you'll be able to build a projective/injective model structure on $[C^{op},sSet]$ which generally (especially if $C$ is rich enough) will have a "simplicial decompletion" on the category of presheaf of sets of $C$.

So it remains to look at example of such model for $(\infty,n)$-category. But here much work is left to be done (and to be completely honest this is something that I'm very interested in and actively working on) One reason for this is that people have generally tried to come up with small and simple models, while here we ask about very big models.

Here the hope is that $(\infty,n)$-category can be represented by a model structures on presehaves of sets and or space over the category of "plex" (the representable in the category of polygraphs) probably with some "stratification" in the spirit of the complicial model. So far I have been focusing mostly on modeling $\infty$-groupoids for simplicity, but I expect the extention to $\infty$-category will not be the hardest part. Even in the groupoid case I'm stuck for technical reason for general polygraphs, and I can only make the theory works for "regular polygraphs", but this is still a fairly large class (containing $\Theta, \Delta$ and many other things and closed under the Gray tensor products) and I've shown that regular polygraphs form a presheaf category and carries a natural model structure that models all $\infty$-groupoids. I believe extensing this to a model structure on "stratified regular polygraphs" modeling weak $(\infty,n)$-categories should be possible with a bit of work (probably one more 80 pages or so paper to be added to the story...)

So, I think, a very simple, and 'essentially' unique, way to obtain such situation is to construct model categories for $(\infty,n)$-category, which are categories of either presheaf of set or presheaf of spaces over a small category $C$ of 'diagrams'. (You can generally jump from a description as presheaf of sets to a description as presheaf of spaces, using some variant of the simplicial completion techniques as illustrated for example in the case of the category $\Theta$ Here.)

(And of course model that are not exactly presheaves can also give some partial answer to your question as long as they contain some subcategories of cofibrant objects that can be thought of as diagrams.)

While producing a rigorous argument for the converse will be difficult, it definitely feels also true: as soon as you have a rich enough class of diagrams $C$ with the sort of properties you are requiring one should be able to prove some sort of Nerve theorem (an $\infty$-categorical version of monads with arities or Nervous monads) to show that $(\infty,n)$-categories can be represented as presehaves of spaces on $C$ satisfying some segal type conditions, for which you'll be able to build a projective/injective model structure on $[C^{op},sSet]$. This model structure will also often (especially if $C$ is rich enough) have a "simplicial decompletion" on the category of presheaf of sets of $C$.

So it remains to look at examples of such model for $(\infty,n)$-categories... But here much work is left to be done (and to be completely honest this is something that I'm very interested in and actively working on) One reason for this is that people have generally tried to come up with small and simple models, while here we ask about very big models.

Here the hope is that $(\infty,n)$-category can be represented by a model structures on presehaves of sets and or space over the category of "plex" (the representable in the category of polygraphs) probably with some "stratification" in the spirit of the complicial model. So far I have been focusing mostly on modeling $\infty$-groupoids for simplicity, but I expect the extention to $\infty$-category will not be the hardest part. Even in the groupoid case I'm stuck for technical reason for general polygraphs, and I can only make the theory works for "regular polygraphs", but this is still a fairly large class (containing $\Theta, \Delta$ and many other things and closed under the Gray tensor products) and I've shown that regular polygraphs form a presheaf category and carries a natural model structure that models all $\infty$-groupoids. I believe extending this to a model structure on "stratified regular polygraphs" modeling weak $(\infty,n)$-categories should be possible with a bit of work (I mean by that: it is probably a lot of work, but very feasible if someone want to spend some time on it) and this would constitute a very good answer to your question.