I went back to this question a few days ago and found the solution: it is indeed a true model structure.
I have two (related) approaches to this, but anyway the key point is the semi-simplicial approximation theorem:
Theorem: If $A \hookrightarrow B$ is a cofibration (=an inclusion) of semi-simplicial sets, $C$ is a semi-simplicial Kan complex, $f:A \rightarrow C$ is a semi-simplicial map and one has $g: |B| \rightarrow |C|$ a continuous map on the geometric realization that extends $|f|$ then there is a semi-simplicial map $g' : B \rightarrow C$ which extends $f$ and such that $|g'|$ is homotopy equivalent to $g$ relative to $|A|$.
This can be found in the first paper of Rourke and Sanderson on semi-simplicial sets as theorem 5.3
From this one can deduce a rather direct proof that the trivial cofibrations of the model structure mentioned in the question are weak equivalences:
Lemma: A map of algebraic semi-simplicial Kan complexes is an equivalence if and only if it is an equivalence on the geometric realization of the underlying semi-simplicial sets.
Indeed, because of the semi-simplicial approximation mentioned above, the $\pi_n$ of a Kan complex and of its semi-simplicial geometric realization are the same.
Now if $A$ is a semi-simplicial algebraic Kan complex, then gluing a horn inclusion to $A$ is done as follows: first, one glues the horn inclusion on the underlying semi-simplicial set and then one iteratively glues all the horn inclusions that do not already have a chosen filling (and one never needs to collapse anything, since the gluing of horn inclusions are monomorphism). In particular, the final Kan complex $A'$ is obtained from $A$ only by gluing horn inclusions and hence $A \rightarrow A'$ is clearly a weak equivalence on the semi-simplicial geometric realization, which proves the claim.
But one can also see the whole model structure mentioned in the question in a nicer way:
Theorem: There is a 'right' semi-model structure on semi-simplicial sets, whose fibrations and trivial fibrations are the Kan fibrations and Kan trivial fibrations and weak equivalences are the maps that induce weak equivalences on the geometric realizations (cofibrations are the monomorphisms).
I am not sure of the utilization of the word "right" here, what I mean is the dual of the notion mentioned in the question, i.e. all trivial cofibrations are weak equivalences, but only the trivial fibrations of fibrant codomain are weak equivalences.
From this theorem, one concludes by observing that Nikolaus' construction of model categories of algebraically fibrant objects also works for a semi-model category and produces a semi-model category. But if one applies it to a right semi-model category one gets a right semi-model category in which every object is fibrant and this is an ordinary model category.
Proof: One has two weak factorization systems by the small object argument, and weak equivalences obviously satisfies $2$-out-of-$3$ and even $2$-out-of-$6$. Trivial cofibrations are obviously weak equivalences. Using the approximation theorem above, weak equivalences between fibrant objects can be characterized as bijections on the semi-simplicial $\pi_n$, from which one easily deduces that the trivial fibrations between fibrant objects are weak equivalences.
We conclude by proving that fibrations between fibrant objects which are also weak equivalences are trivial fibrations:
Let $f :X \rightarrow Y$ be a fibration and a weak equivalence, let $a : \partial \Delta_n \rightarrow X$ be a boundary of $X$ and $b : \Delta_n \rightarrow Y$ be a filling of $f(a)$ in $Y$. As $f$ is an equivalence on the geometric realizations, $a$ has a filling on the geometric realization. Morevoer, as $Y$ is fibrant, there is a map $a' : \Delta_n \rightarrow X$ such that $f(a')$ is homotopy equivalent to $b$ relative to the boundary.
The point is then that with a bit of work on simplicial combinatorics one can construct a semi-simplicial set $I$ such that: $I$ contains $\Delta_n \coprod_{\partial \Delta_n} \Delta_n$ as a sub-complex, the map $\Delta_n \hookrightarrow I$ is a trivial cofibration and the geometric realization of $I$ is (at least up to homotopy) $\Delta_n \coprod_{\partial \Delta_n} \Delta_n$ with a filled interior.
At this point, one has a map from $I$ to $Y$ at the level of geometric realizations that extends our already existing map from $\Delta_n \coprod_{\partial \Delta_n} \Delta_n$. Hence by the approximation theorem one has such a map as a semi-simplicial map, the inclusion of $\Delta_n$ into $I$ is a trivial cofibration and therefore $I$ can be lifted to $X$ and the other inclusion of $\Delta_n$ into $I$ provides our lifting. This concludes the proof of the theorem.
