If you don't mind, I'll talk about strinct $\infty$-categories, but weak $(\infty,n)$-category to avoid discussing the 'problem' regarding the non uniqueness of the meaning of $(\infty,\infty)$-categories mentioned [Here][1].

Also I don't think what follows completely answer the question, as Harry said in the comment, this is a quite open problem.


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][2].)

Obviously, if you have such a model then the objects of your category $C$ have all the properties you expect. Also notes that generally presheaf over $C$, or at least some presheaves over $C$, still corresponds to some kind of diagrams and so you get this sort of construction not justs for objects of $C$ but also for more general diagrams build out of objects of $C$. (basically, the cofibrant object of your model structure).


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][3] or [Nervous monads][4]) 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.

The first that comes to mind is obviously the category $\Theta_n$ (see for example Dimitri's paper mentioned above) which obviously fits into this picture, and Verity [complicial set models][5] which does the job for Street Orientals, with the only problem that this model hasn't been compared to others ones.


But maybe the most interesting example already worked out regarding your question is the so-called "Colossal model" constructed in Barwick and Schommer-Pries' paper [On the unicity of the theory of higher categories][6], which does this for a large class of gaunt categories. But I'll have to read again this paper before I can say something more precise here.


Finally my own work on polygraphs and the Simpson conjecture ([here][7] and [enter link description here][8]) is basicaly an attempt to prove this result for the very large class of all "non-unital [polygraphs][9]".

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...)



Also notes that in everything I have discussed above the functor "$F$" of your question is not really expecte to be fully faithful. I don't know if this is a requirement you have or if you are happy staying with "polygraphic morphisms".


  [1]: https://mathoverflow.net/questions/134057/is-there-an-accepted-definition-of-infty-infty-category/134099#134099
  [2]: https://arxiv.org/pdf/1206.4354.pdf
  [3]: https://arxiv.org/pdf/1101.3064.pdf
  [4]: https://arxiv.org/abs/1805.04346
  [5]: https://arxiv.org/abs/math/0604414
  [6]: https://arxiv.org/pdf/1112.0040.pdf
  [7]: https://arxiv.org/abs/1711.00744
  [8]: https://arxiv.org/abs/1807.02627
  [9]: https://ncatlab.org/nlab/show/computad