The way I understand the electron density of the s-orbital is that there is a probability of finding an electron in the nucleus, but we have to remember what the electron density actually is, the probability per point in space. When we consider the relative number of points in space (i.e. the volume) occupied by the nucleus compared to the total volume occupied by the electron we realise the although the probability of finding the electron is high at the nucleus compared to a point is space outside, there are trillions more points in space outside the nucleus than inside, hence why electrons don't wind up in the nucleus that often. Sorry that's a bit waffly.
Electrons do go into the nucleus and under specific circumstances can be captured by a proton to form a neutron and emit a neutrino and reduce the atomic number by 1, so called "K-shell capture" a form of radioactive decay.