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.
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 depending only on the scale $\varepsilon$. This can be done since Delaunay nets are constructed greedily. Then use some finite simplexwise procedure to interpolate between the original boundary and the new, subdivided boundary.
Now take a PL manifold $M^d$ equipped with an arbitrary PL triangulation.
- 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 $C(d)$.
- 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 $C(d,2)$.
- 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 $C(d,3)$.
- 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.