I believe the following seem to be a very simple counterexample without it being related to Whitehead obstruction as suggested by Tyler Lawson.
Consider the simplicial sets $D$ freely generated by:
It is contractible: indeed it is connected, Its $\pi_1$ is generated by $t$ subject to the relation $(\gamma) : t^2=t$ hence is trivial. Its homology is easy to compute and is trivial.
The inclusion of $\Delta$ into $D$ is hence a cofibration and a weak equivalence, i.e. a trivial cofibration.
$D$ has only three non-degenerate cells, and taking a pushout by a Horn inclusion $\Lambda^k[n] \hookrightarrow \Delta[n]$ always add exactly two non-degenerate cells of dimension $(n-1)$ and $(n)$.
So if $\Delta \hookrightarrow D$ were a pushout of Horn inclusion it would have to be a pushout by $\Lambda^i \hookrightarrow \Delta$.
Now for each $i$ there is only one maps $\Lambda^i \rightarrow \Delta$ along which to take the pushout and none of them gives $D$.
Here is an attempt to get a counter example in the case of the Joyal model structure.
I'm constructing $D'$ by adding to $D$ the following cells:
- a $1$-cell $u:x \rightarrow x$.
- a $2$-cell $\theta$ such that $d_0 \theta =t$, $d_2 \theta = u$ and $d_1 \theta$ is the unique degenerate $1$-cell.
$D'$ is contractible for the Joyal model structure: its homotopy category is freely generated by an idempotent with a left inverse, hence is trivial. Hence $D'$ is an $\infty$-groupoids with trivial $\pi_1$ and the computation of its homology shows that it is contractible.
If it were obtained from $\Delta$ as pushout of Inner Horn inclusion the argument of number of non-degenerate cells shows that it has to be a pushout of to copies of $\Lambda^1 \rightarrow \Delta$.
But if that were the case, $t$ would have to be added as pushout of $\Lambda^1 \rightarrow \Delta$ at some points, in particular $t$ would appears as $d_1 y$ for $y$ a non-degenerate two cell, with $d_0 y$ and $d_2 y$ not equal to $t$. but there are only two non-degenerate two cells and none of them has this property.