Stimpson J. Cat said:
In the lowest energy state there is a nonvanishing probability of the electron being found arbitrarily close to the nucleus. But what does this mean?
We all have an intuitive idea of particles being little solid balls of matter, but this is not correct. The electron does not have any "size" as such. It is not meaningful to talk about it actually "striking" the nucleus. In fact, you could find it arbitrarily close to the nucleus, but then a short time later, back far away from it.
Your question seems to imply that you think that the electron should somehow be captured by the nucleus. The thing is that in order for this to happen, some sort of reaction between the electron and the nucleus must be possible. The simplest such reaction is for the electron and proton to combine to form a neutron, and omit a neutrino. The thing is that for stable nuclei, this reaction will not occur, because the total energy after the reaction is considerably more than before the reaction. It is possible for unstable nuclei, though. This is known as K capture.
Remember that in QM, it is not sufficient, or even necessary, for two particles to "touch" for a reaction to occur. In fact, it is not even really meaningful to talk about them touching. What is necessary is that a reaction be possible, and then the probability of such a reaction depends on the relative energy before and after the reaction, and the proximity of the wave-functions of the particles. This is why in QM, scientists refer to the "cross-section" of an interaction, rather than talking about "probabilities of hitting". The cross-section is in effect a direct measure of the likelihood of the reaction occurring.
