@user62782 If I understand correctly this functor sends a simplicial set to the discrete cubical set with the same vertices. I would assume that this is not what you wanted.

I'm not sure if this really addresses the question but it sounds like Local and Stable Homological Algebra in Grothendieck Model Categories by Cisinski and Déglise might be relevant.

The distinction between classical operads and cartesian operads that I describe in my answer is that in algebraic theories presented with operads you cannot insert "dummy variables". This means that you will not be able to identify affine combinations like $a x + b y$ and $a x + b y + 0 z$ since all identities in an operad need to have exactly the same set of variables on both sides.

By analyzing pushout products of boundary inclusions you can check that the lifting property that I described is equivalent to the (seemingly stronger) condition that $\mathcal{C}^{\Delta[m]} \to \mathcal{C}^{\partial\Delta[m]}$ is an acyclic Kan fibration for $m \ge n + 2$. This is invariant under categorical equivalences.

The strict definition of $(n, 1)$-categories is indeed not invariant under categorical equivalences. If you want to use it to construct a Bousfield localization from it, what model structure would you localize? I don't see any candidate other than the Joyal model structure and if you try that with the Joyal model structure you will end up with the same localization that I described anyway.

Why do you believe that the lifting property I gave is not equivalent to $(n - 1)$-truncated mapping spaces? It is fairly easy to see, e.g., you can assume that your quasicategory is the coherent nerve of a fibrant simplicial category and use the left adjoint $\mathfrak{C}$.

I have just remembered that Rezk considers the Segal space version of this problem in A Cartesian Presentation of Weak n-categories. In fact, he covers general $(n, k)$-categories not just $(n, 1)$-categories. He uses a similar Bousfield localization of the model structure for complete $\Theta_k$-spaces.