Indeed the points do bounce all over the unit interval, though they more often bounce near one of the ends.
Does that distribution look familiar? You might recognize it from Bayesian statistics. It’s a beta distribution. It’s symmetric, so the two beta distribution parameters are equal. There’s a vertical asymptote on each end, so the parameters are less than 1. In fact, it’s a beta(1/2, 1/2) distribution. It comes up, for example, as the Jeffreys prior for Bernoulli trials.
The graph below adds the beta(1/2, 1/2) density to the histogram to show how well it fits.
It’s an interesting bit of math and statistics, and John provides some Python demo code at the end.