Regarding the discussion we had some time ago, that simplicially enriched categories don't admit a simplicial model structure (or are at least not known to), does this pose any problem in making the statement "the nerve of the Bergner model structure on simplicial categories is equivalent to the nerve of the category of quasicategories?"

Ah, I guess when you take the Hammock localization of the category of simplicial categories, with the Bergner model structure, you get a simplicial category, but not a simplicial model category. But that's okay, because (I think?) the simplicial nerve of this simplicial category will be equivalent to the "$\infty$-category of $\infty$-categories," since the category of simplicial sets with the Bergner model structure is Quillen equivalent to simplicial sets with the Joyal model structure.

@JonathanBeardsley If that is the case that they don't admit a simplicial model structure, I think you need all of Tim's statement

because the simplicial structure on sCat is modeling an infty,2 cat

while its model structure models an infty,1 cat

Somehow knowing that sCat and sset_Joyal are QEQ by the HC-nerve is not essential to the theory, only that the HC-nerve is a Quillen adjunction

probably overgeneralizing herw, but I cant think of a single theorem that relies on this fact