This has not been done, and there are good reasons for it: While $sSet$-enriched categories are indeed very good to easily get examples of $\infty$-categories, they are very bad at understanding what are functors between $\infty$-categories. The problem is that a "sSet-enriched functors" is a way too strict notion, that need to satisfies a very strong condition of compatibility with composition, while the correct definition of functor between $\infty$-categories should only satisfy this condition up to equivalence.

When you work with quasi-categories or Complete Segal spaces, $\infty$-functors are justs the morphisms of simplicial sets or bisimplicial sets.

For example, to answer your third question: sSet-enriched adjunction definitely induce adjunction between the associated $\infty$-categories, but not all such adjunction will be of this form (for fixed simplicially enriched categories).

The reason while we can use sSet-enriched categories to model $(\infty,1)$-categories despite this problem is because there are enough "nice" sSet-enriched category that have the properties that any "$\infty$-functors" (or "weak functor", whatever this should mean) out of them is equivalent to one that is a sSet-enriched functor (this is a kind of stratification result if you'd like). These "nice" sSet-enriched categories are essentially the cofibrant objects of the Bergner model structure.

So if one wanted to get some sort of nice "synthetic" theory that applies to sSet-enriched categories, or just a setting where one could develop $(\infty,1)$-category theory nicely, then one would need to either:

• Move to a category of sSet-enriched categories and "weak functors" between them (for some definition of weak functor), but this is a fairly complicated category, with poor category theoretic properties. So not a very nice place to work in. It is much simpler to consider that a good way to define these weak functors is as morphisms between the associated quasi-categories, or between some associated complete Segal spaces, and works in these categories instead.

• Keep taking "cofibrant replacement" all the time everywhere in the theory. This is surely possible, but it makes everything much more complicated and given that we have model where this is not needed (quasi-category, complete Segal spaces, etc...) nobody bothered to develop such a theory. In addition doing so defeat the initial motivation you talk about : Most examples of these interesting sSet-enriched category you are mentioning. And the cofibrant replacement construction is not a very explicit construction - much more complicated that the functor from sSet-enriched categories to quasi-categories or complete Segal Spaces.

As mentioned by Zhen Lin, in both case, this makes the construction of "functors categories" extremely annoying, and this is probably the single most important construction in basic higher category theory.

This has not been done, and there are good reasons for it: While $sSet$-enriched categories are indeed very good to easily get examples of $\infty$-categories, they are very bad at understanding what are functors between $\infty$-categories. The problem is that a "sSet-enriched functors" is a way too strict notion, that need to satisfies a very strong condition of compatibility with composition, while the correct definition of functor between $\infty$-categories should only satisfy this condition up to equivalence.

When you work with quasi-categories or Complete Segal spaces, $\infty$-functors are justs the morphisms of simplicial sets or bisimplicial sets.

For example, to answer your third question: sSet-enriched adjunction definitely induce adjunction between the associated $\infty$-categories, but not all such adjunction will be of this form (for fixed simplicially enriched categories).

The reason while we can use sSet-enriched categories to model $(\infty,1)$-categories despite this problem is because there are enough "nice" sSet-enriched category that have the properties that any "$\infty$-functors" (or "weak functor", whatever this should mean) out of them is equivalent to one that is a sSet-enriched functor (this is a kind of stratification result if you'd like). These "nice" sSet-enriched categories are essentially the cofibrant objects of the Bergner model structure. And "enough" mean that one can always take cofibrant replacement.

So if one wanted to get some sort of nice "synthetic" theory that applies to sSet-enriched categories, or just a setting where one could develop $(\infty,1)$-category theory nicely, then one would need to either:

• Move to a category of sSet-enriched categories and "weak functors" between them (for some definition of weak functor), but this is a fairly complicated category, with poor category theoretic properties. So not a very nice place to work in. It is much simpler to consider that a good way to define these weak functors is as morphisms between the associated quasi-categories, or between some associated complete Segal spaces, and works in these categories instead.

• Keep taking "cofibrant replacement" all the time everywhere in the theory. This is surely possible, but it makes everything much more complicated and given that we have model where this is not needed (quasi-category, complete Segal spaces, etc...) nobody bothered to develop such a theory. In addition doing so defeat the initial motivation you talk about : Most examples of these interesting sSet-enriched category you are mentioning. And the cofibrant replacement construction is not a very explicit construction - much more complicated that the functor from sSet-enriched categories to quasi-categories or complete Segal Spaces.

As mentioned by Zhen Lin, in both case, this makes the construction of "functors categories" extremely annoying, and this is probably the single most important construction in basic higher category theory.