This is just a long comment, which does not really address the main question.
Instead, I give few examples where fibrations are an additional rigidity structure allowing "homotopical commutativity".

In all of the simplicial models for $(\infty, 1)$-categories that you listed, only the fibrant objects of the respective model category structure behave as "categories weakly enriched in $\infty$-groupoids".
Inside this categorical framework (if you want, formal 2-categorical), there is a notion of localisation $C[W^{-1}]$ of an $\infty$-category $C$ with respect to a sub-$\infty$-category $W$. Given a relative category $(C, W)$, it is a theorem that the $\infty$-localisation $N(C)[N(W)^{-1}]$ models the $\infty$-category associated to $(C, W)$ (see for instance the appendix of Brown categories and bicategories of Geoffroy Horel).
However if $(C, W)$ is not fibrant, nor will be its Rezk's relative nerve, i.e. $N^R(C, W)$ is not really an $\infty$-category.

Suppose now your relative category $(C, W)$ has an additional structure of fibrantions that interacts gently with the weak equivalences (for instance, a model category), let's call such a structure a *fibration category*. Lennart Meier showed in Fibration categories are fibrant relative categories that any such relative category is fibrant in the Barwick-Kan model category structure and thus $N^R(C, W)$ *is* an $\infty$-category modelling the $\infty$-categorical localisation $N(C)[N(W)^{-1}]$.

This is just the reflection of something happening entirely in the $\infty$-categorical world. Indeed, in Chapter 7 of Higher categories and homotopical algebra Denis-Charles Cisinski studies the $\infty$-localisation theory of *fibration $\infty$-categories*.
Within this framework, we have the beautiful feature that the $\infty$-localisation of a fibration $\infty$-category has finite $\infty$-limits (see Section 7.5).
Moreover the identification of homotopy limits and $\infty$-limits can be stated as an elegant theorem about an equivalence of $\infty$-categories (see Theorem 7.9.8 and Remark 7.9.10).

The longer story tells that fibration $\infty$-categories are closed by exponentiation, that is if $(C, W)$ is a fibration $\infty$-category and $I$ is an $\infty$-category, then $\underline{\text{Hom}}(I, C)$ is still a fibration $\infty$-category, with fiber-wise weak equivalences $W_I$. If $(C, W)$ is just (the nerve of) a regular fibration category and $I$ (the nerve of) a small category, then the $\infty$-localisation $\text{Fun}(I, C)[W_I^{-1}]$ is equivalent to the $\infty$-functor category $\underline{\text{Hom}}(I, C[W^{-1}])$. (Notice that $(\text{Fun}(I, C), W_I)$ is a fibrant relative category!)

In the case where the fibration $\infty$-category is actually a model $\infty$-category, the theory has been studied by Aaron Mazel-Gee, as Dmitri Pavlov says in his nice answer.