I've been thinking about this question too! It seems that showing that the Reedy and injective model structures agree for presheaves on $R$ boils down to having a well-behaved notion of "degeneracy".

Suppose you have a simplicial set $X$. Given an $n$-simplex $x\in X([n])$, you say it is *degenerate* if there exists a $k<n$, a $k$-simplex $y\in X([k])$, and a surjective map $\sigma : [n]\to [k]$ such that $X(\sigma)(y)=x$. The simplex $x$ is *non-degenerate* if it is not degenerate.

Two facts:

If $X\subseteq Y$ is an inclusion, and $x\in X([n])$ is non-degenerate as a simplex of $X$, then it is also non-degenerate as a simplex of $Y$.

For every $x\in X([n])$, there exists a *unique* pair $(y,\sigma)$ consisting of a surjection $\sigma:[n]\to [k]$ and a non-degenerate $y\in X([k])$ such that $X(\sigma)(y)=x$.

These facts turn out to be what you need to show that the Reedy and injective model structures coincide on $\mathrm{sSet}^{\Delta^{op}}$; in particular, you can show that all monomorphisms are Reedy cofibrations (which is the hard part). If you replace $\Delta$ with a Reedy category $R$ and make the appropriate definition of "degenerate" and "non-degenerate", conditions 1 and 2 are sufficient for injective=Reedy.

This leaves the question: why are 1 and 2 true for simplicial sets? I've known these facts for years, but only recently realized that I didn't know the proof! And then when I constructed a proof on my own, it turned out to be quite ugly. The good news is that there is in fact a very nice proof, due to Eilenberg and Zilber, and which you can find in section II.3 of Gabriel-Zisman, "Calculus of Fractions and Homotopy Theory".

I think these ideas generalize to $\Theta_n$.