Why should the statistical value of life exist?
You asked: 


  why should there exist a single value of $X$ that satisfies this definition for all values of $p$, or even all values of $p$ that are sufficiently close to $0$


There isn't such a value. I would hope that no one claims that there is.

The statistical value of life is a (somewhat lazy) calculation of convenience. Lots of business case protocols need a value for anything that's going into the business case. Changing the probabilities of survival is an outcome of many interventions for which the decision-makers have insisted on business cases, so some method is needed to value these probabilities.

One of the earliest ways to do this, back when relevant research was scarcer than it is today, and computational power was much more limited, was to assign a single value of life, which was calculated using methods that assumed a priori that there existed a single value of $X$ that was an adequate approximation for all values of $p$ that are sufficiently close to $0$.

That method is still used today largely due to institutional inertia.