@JamesMartin Yes, that's a very good point, and possibly a fatal one. The tickle in my brain is as follows: Let $X$ be (the realisation of) a Mátern hard-core point process. Perhaps if $Y$ is a very "aggressive" thinning of $X$ then it will be a good approximation to a homogenous Poisson process, becoming perfect in the limit? So yes, there are no points within distance $r$, but on the big scale the $r$ looks like it is tending to zero?

The question that was posed has an intuitive interpretation, "Suppose that $p(x)$ represents the density of the material at any point $x$. If changes in density are 'nice' throughout $X$, is it true that shortest paths are smooth and of bounded curvature?" The intuition is "yes" if $p(x)$ is smooth.