This is a long comment. Simon's answer is correct, but there are other ways around the difficulty of forming the right internal Hom.
If $C$ is a full simplicial subcategory of a simplicial model category such that all objects of $B$ are fibrant and which is stable under the factorization of maps into trivial cofibration/fibration, then, the formation of the simplicial category of simplicial functors $Fun(A,C)$ for each simplicial category $A$ is compatible with Dwyer-Kan equivalences. In particular, there is no need any more to to take cofibrant resolutions of $A$. In particular, on may consider a simplicial category $B$ and take for $C$ the full simplicial subcategory of simplicial functors $B^{op}\to Kan$ with values in Kan complexes which are weakly equivalent to those of the form $Hom_B(-,b)$ for some object $b$ of $B$. In other words, we have a perfectly fine access to the $(\infty,1)$-category-theoretic Yoneda embedding in the context of simplicial categories, and it fits very well with its more rigid sSet-enriched counterpart. One can also express the theory of Kan extensions very naturally in the setting of sSet-enriched categories. In other words, if we think of objects of an $(\infty,1)$-category as representable functors, given two $(\infty,1)$-categories $A$ and $B$, we can associate their presheaf categories $\widehat A$ and $\widehat B$, respectively, and think of functors $A\to B$ as colimit preserving functors $\widehat A\to\widehat B$ that preserve representable presheaves. This point of view is very robust and has a nice interpretation in all models of $(\infty,1)$-categories, including sSet-categories. I would like to insist on the fact that this is the way the theory has been developed by Dwyer and Kan (all the assertions I made above without proof are in their papers). I would not qualify this approach as "annoying" - it is quite efficient, actually. In fact this point of view is very robust and also applies to categories enriched in a (nice enough) model category; for instance, in the case of chain complexes, this leads to Bertrand Toen's paper The homotopy theory of dg-categories and derived Morita theory which gives a very workable approach (197 citations on Mathscinet).
Having access to the Yoneda embedding is indeed the central feature of $(\infty,1)$-category theory: this encodes straightening/unstraightening together with the theory of (pointwise) Kan extensions, out of which on can do everything else (abstract constructions as well as computations). Indeed, given a functor $p:X\to A$, we compose $p$ with the Yoneda embedding and take the colimit: this is "straightening". "Unstraightening" is the claim that "straightening" is a left Boudfield localization whose right adjoint takes values in cartesian fibrations whose fibers are $\infty$-groupoids. We could summarize that, in other models of $(\infty,1)$-categories (such as quasi-categories or complete Segal spaces), there is a nice theory (which looks like a literal interpretation the language of ordinary category theory) where the main difficulty is the construction of the universe (the $\infty$-category of $\infty$-groupoids), whereas in the model of simplicial categories, there is a rather easy access to the universe (Kan complexes naturally form a simplicial category) and to pointwise Kan extensions, but everything else is mode difficult. Lurie's take in Higher Topos Theory is to take advantage of both approaches to begin with.
A final comment about the approach of weak functors: if we weaken the notion of functor, we should weaken everyting, including the composition law. A way to do it is to weaken the notion of finite sequence of composable functors of length $n$, such as $A_0\to A_1\to\ldots\to A_n$, for each $n\geq 0$. The fun part is that we do not obtain a category, but a quasi-category of simplicial categories (we could do that by replacing categories with algebras over any coloured operad). This is how Boardmann and Vogt discovered the very notion of quasi-category (that they coined as weak Kan complexes). In other words, the first example of quasi-category that is not the nerve of a $1$-category comes from this idea of weakening!
