No, there need not be such a geometric triangulation. Here is a construction.
Consider $A$, a flat annulus, of width $W$ and length $L$. Here we assume that $W$ is very large and $L$ is very small. (That is, take a $W$ by $L$ rectangle and glue the long sides.)
Let $\alpha$ and $\beta$ be the components of $\partial A$. We glue many sub-intervals in $\alpha$ to isometric sub-intervals in $\beta$, via some complicated permutation. This gives a high genus surface $S$. The singular points all live in a graph -- namely the image of $\partial A$ after taking the quotient. (In fancier language: $S$ is the suspension of an interval exchange transformation.)
Note that the diameter of $S$ is at most $(W+L)/2$. However, any geometric triangulation must have at least two edges crossing $A$ and these edges have length at least $W$.