SHORT ANSWER: Median age is not simply better than mean age; however, you may have noticed that more people use it. So a better question might be: "Why would more demographers use median age than mean age?"
A statistic, as a vocabulary term, has its origins in the state (nominally a legal entity) attempting to understand its human population. So think about the people in those governments and how much information they want or need, as well as how much time they have to devote to understanding the precise mathematical meaning of scientific words.
The easiest way to sum up a lot of data, without using a picture, is to report a single number; this is known as an estimator for the parameter in question (in this cage, the time elapsed since birth of a human being, precise to the level of years). As E.T. Jaynes showed in his book Probability Theory: The Logic of Science, one could choose to construct an estimator based on a utilitarian loss function summarizing the consequences of making a mistake based on using a single number instead of an entire data set when making decisions based on that information.
In Jaynes' book, he shows via mathematical proof that the mode, or maximum likelihood estimator, is the estimator minimizing loss shaped like a Dirac delta function. The mean minimizes quadratic loss functions, such that the further one gets from the estimate, the quantity of loss (undesirable consequence) goes up very quickly once you pass the unit scale.
The median, on the other hand, minimizes a loss function shaped like an inverted triangle, such that it is only five times less desirable to be off by one unit of precision, rather than 25 times (as in the case when using the mean). In fact, the unit of precision makes no difference whatsoever, because there is no curvature in such a triangular pointy loss function.
With this theoretical foundation, one could literally draw loss functions that are not symmetric at all and form an infinite number of new estimators custom-suited to the needs of their consumers/users. Another alternative to dealing with the cultural expectation of a single number is to educate those same users/consumers of information that a measure of central tendency can provide more information when paired with other parameters of a distribution, such as variance, skew, and kurtosis (might want to start with just variance and skew to ease them into it.
The variance is just one example of a dispersion measure; another Jaynes suggests (in other writings) is to form a Bayesian posterior distribution and calculate the width of the shortest credible interval with value 0.5 (or confidence interval/standard deviation etc. if you do not buy into Bayesian theory--please, let's not get sidetracked). A more intuitive method which is possibly easier for more people to grasp would be the inter-quartile range, especially when reported with the median as its corresponding measure of central tendency.
I am not sure if there is a non-parametric form of skew or kurtosis, but if they do exist they will almost certainly be easier to understand than these parametric analogs. I have a hunch that a major, if not dominant, part of the reason median age crops up more often than mean age is because it simply appeals more to people with less time or desire to go into theoretical details about things like sigma-algebras, Lebesgue measure theory, etc. which are all technically necessary to understand the more common foundations of probabilistic reasoning.