I know that the references usually regarded as standard for simplicial localizations are the Dwyer and Kan's three articles from the 80's. I would be interested in a more modern approach to the subject, and in particular the following two issues keep surfacing in my work, for which I'd appreciate to be able to cite something that is more compact and student-friendly than the original works.

1. Define the $\infty$-category $L(\mathcal{C}, \mathcal{W})$ associated to a relative category $(\mathcal{C}, \mathcal{W})$ as the homotopy pushout of the span $\coprod_{\mathcal{W}} J \leftarrow \coprod_{\mathcal{W}} \Delta^1 \rightarrow N(\mathcal{C})$ in $\textbf{sSet}$ (or anything equivalent to it). I would like to prove that for a simplicial model category $\underline{\mathbf{M}}$ then there is an equivalence of $\infty$-categories $$ \text{N}_{\Delta}(\underline{\mathbf{M}}^{cf}) \simeq L(\mathbf{M}, \mathcal{W}) $$ where $\mathbf{M}$ is the underlying (unenriched) category of $\underline{\mathbf{M}}$. Ideally, I would like to avoid the three above mentioned articles, preferring instead later developped and thoroughly studied technologies such as the equivalence between Joyal and Bergner model structures on $\mathbf{sSet}$ and $\mathbf{sCat}$, or that between Barwick-Kan and Rezk model structures on $\mathbf{RelCat}$ and $\mathbf{ssSet}$, or the like.

2. I suspect that, for an arbitrary relative category, there should be an equivalence $$\text{N}_{\Delta}(L^H(\mathcal{C}, \mathcal{W})) \simeq L(\mathcal{C}, \mathcal{W})$$ where $L^H$ denotes the hammock localization, and $L$ is the associated $\infty$-category as defined above, but I can't find a proof of this. Again, a modern approach using high technology is preferable to the original paper where the hammock localization was introduced.

I know that the references usually regarded as standard for simplicial localizations are the Dwyer and Kan's three articles from the 80's. I would be interested in a more modern approach to the subject, and in particular the following two issues keep surfacing in my work, for which I'd appreciate to be able to cite something that is more compact and student-friendly than the original works.

1. Define the $\infty$-category $L(\mathcal{C}, \mathcal{W})$ associated to a relative category $(\mathcal{C}, \mathcal{W})$ as the homotopy pushout of the span $\coprod_{\mathcal{W}} J \leftarrow \coprod_{\mathcal{W}} \Delta^1 \rightarrow N(\mathcal{C})$ in $\textbf{sSet}$ (or anything equivalent to it). I would like to prove that for a simplicial model category $\underline{\mathbf{M}}$ then there is an equivalence of $\infty$-categories $$ \text{N}_{\Delta}(\underline{\mathbf{M}}^{cf}) \simeq L(\mathbf{M}, \mathcal{W}) $$ where $\mathbf{M}$ is the underlying (unenriched) category of $\underline{\mathbf{M}}$. Ideally, I would like to avoid the three above mentioned articles, preferring instead later developed and thoroughly studied technologies such as the equivalence between Joyal and Bergner model structures on $\mathbf{sSet}$ and $\mathbf{sCat}$, or that between Barwick-Kan and Rezk model structures on $\mathbf{RelCat}$ and $\mathbf{ssSet}$, or the like.

2. I suspect that, for an arbitrary relative category, there should be an equivalence $$\text{N}_{\Delta}(L^H(\mathcal{C}, \mathcal{W})) \simeq L(\mathcal{C}, \mathcal{W})$$ where $L^H$ denotes the hammock localization, and $L$ is the associated $\infty$-category as defined above, but I can't find a proof of this. Again, a modern approach using high technology is preferable to the original paper where the hammock localization was introduced.