Combinatorial proof that some model categories are monoidal/enriched?

I'll do you one better: you don't need a generating set of acyclic cofibrations. You just need what Simpson calls a pseudo-generating set, i.e. a set of acyclic cofibrations which suffices to detect fibrant objects and fibrations between fibrant objects (i.e. an object is fibrant iff it lifts against every morphism in your pseudo-generating set, and similarly for a map between fibrant objects being a fibration).

That is, I claim your observation still holds if you only check with the morphisms of a pseudo-generating set rather than an actual generating set of acyclic cofibrations.

Explicit generating sets of acyclic cofiberations are hard to find, but pseudo-generating sets abound. For example, the inner horns plus the endcap inclusion of the walking isomsomorphism form a pseudo-generating set for the Joyal model structure. A great source of them comes from Cisinski-Olschok theory: their construction $S \mapsto \Lambda(S)$ is a minimal way of getting a pseudo-generating set out of a set of morphisms. Because the construction of $\Lambda(S)$ is explicit and combinatorial, one can often show that $\Lambda(S)$ is in the cofibrant closure of $S$, so that $S$ was already a pseudo-generating set (and showing this is an essentially combinatorial exercise). For example, this works in the case of the Joyal model structure.

My impression is that everything that Cisinski/Olschok do is very explicit, and is probably constructive.

This is far from a comprehensive answer, but since you ask for references, here are two.

The result for simplicial sets is obtained by an explicit computation in Appendix H of Joyal's The Theory of Quasi-categories and Its Applications.

For $\Theta_2$-sets, there is Chapter 3 of Oury's Duality for Joyal's Category $\Theta$ and Homotopy Concepts for $\Theta_2$-sets.