It seems to me that by the algebraic Riemann Hilbert functor(which factors through the analytification map) and also the analytic Riemann Hilbert functor that the (derived) category of (algebraic) regular holonomic D-modules embeds into the (derived) category of analytic regular holonomic D-modules. Is this correct? I want to make sure I'm not missing anything. Is it also correct to say that this is a nontrivial fact and needs much of the theory of Riemann Hilbert correspondence to prove?

Yes, for a smooth algebraic variety $X$, the analytification functor

$D^b(\mathcal D_X)_{rh} \to D^b(\mathcal D_{X^{an}})_{rh}$

is fully faithful.

As you note, it is a consequence of the usual algebraic and analytic versions of the Riemann-Hilbert correspondence for $D$-modules: analytic $D$-modules correspond to complexes of sheaves which are constructible with respect to analytic stratifications and the algebraic category corresponds to the subcategory for which the stratification is algebraic. This is stated, for example, as Proposition 7.8 in this paper of Brylinski.

As to how much of the theory of Riemann-Hilbert one needs to prove this fact, here are my thoughts.

One way to prove the algebraic version of Riemann-Hilbert is by first proving the analytic version then proving the fully faithfulness of analytification statement in your question. If I understand correctly this is the approach taken in Brylinski's paper (there is a comment to that effect on page 174 of the book by Hotta-Tanisaki-Takeuchi). So in some sense, the statement in your question seems kind of orthogonal to the analytic version of the Riemann-Hilbert correspondence at least.

To prove the fully faithfullness of analytification, first consider the case where $X$ is projective, in which case it follows from a suitable application of GAGA theorems (see Theorem 7.1 in that paper of Brylinski). Now, for a general smooth algebraic variety $X$, consider a nice compactification $j:X \hookrightarrow \overline{X}$. Then for a r.h. algebraic D-module $M$ on $X$ we can look at the $D$-modules $j_\ast M$ which will be r.h. on $\overline{X}$ thus reducing to the projective case (note that in the algebraic setting, unlike the analytic, the definition of regularity includes being regular "at $\infty$").

This proof requires some theory of regular singularities, but is independent of certain other fundamental facts involved in RH. For example, there is the basic fact that an analytic flat connection on $X$ extends uniquely to a meromorphic connection on $\overline{X}$ with regular singularities (I think this is due to Deligne?). There is also the constructibility of the solution/de Rham complex. I would say these essential aspects of Riemann-Hilbert are kind of orthogonal to (or at least independent of) the comparison of algebraic and analytic theories in your question.