Here's a stab at a proof that works for non-smoothable manifolds, which seems to be related to Florian Frick's sketch.  I'm going to call manifolds with at most $L$ simplices incident to a vertex "of geometry bounded by $L$" or just "of bounded geometry".  The operator $*$ denotes the [join][1].

First: **Lemma.** Every PL sphere of geometry bounded by $L$ can be filled with a PL ball of geometry bounded by $C(L)$.

*Proof of lemma.* Use the Delaunay triangulation argument in Ian Agol's answer.  Smooth out the sphere, take a Riemannian filling, and take a Delaunay triangulation of the filling such that the Delaunay triangulation on the boundary is a subdivision of the original triangulation.  You can use a specific sequence of subdivisions of each simplex $\Delta^i$ depending only on the scale $\varepsilon$.  This can be done since Delaunay nets are constructed greedily.  Then use some fixed extensions of these subdivisions to $\Delta^i \times [0,1]$ to interpolate between the original boundary and the new, subdivided boundary.

Now take a PL manifold $M^d$ equipped with an arbitrary PL triangulation.

1. Barycentrically subdivide once.  The resulting complex has a vertex for each face of $M$.  The link of the vertex corresponding to a $k$-face $\sigma$ looks like $\partial\Delta^{d-k} * \operatorname{BarySub}(\operatorname{link}(\sigma))$.  The star of the vertex corresponding to every $d$-face and $(d-1)$-face already has geometry bounded by $C(d)$.
2. Now look at the star of each vertex corresponding to a $(d-2)$-face.  This looks like $\partial\Delta^{d-2} * CS$ where $S$ is a circle with many edges.  Using the lemma, we can replace the interior of this subcomplex by $\partial\Delta^{d-2} * D(S)$, where $D(S)$ is a bounded geometry filling of $S$.  Now the subcomplex $D(S)$ has geometry bounded by $C(d,2)$.
3. If we now look at the star of each vertex corresponding to a $(d-3)$-face, it looks like $\partial\Delta^{d-3} * C\Sigma$ where $\Sigma$ is a triangulated 2-sphere.  This surface is a union of patches which are the $D(S)$'s from the previous step, therefore it has bounded geometry.  Using the lemma, we can replace $C\Sigma$ with a 3-ball of geometry bounded by $C(d,3)$.
4. Keep going like this.  At the $k$th stage, the $k$-skeleton of the dual CW complex to the original triangulation is subdivided into a triangulation with geometry bounded by $C(d,k)$.  At the end of the procedure, the resulting manifold has bounded geometry.

  [1]: https://en.wikipedia.org/wiki/Join_(topology)