Why doesn't every pure enantiomer racemise?
Because the activation energy for the enantiomeric H-Atom to jump to the other side of the carbon atom's sp3 tetraeder (by whatever mechanism) is so high that it is just not available.

You're right, statistically, every energy is available (with very bad odds). But very likely other processes, and more destructive ones, have a lower activation energy. Your compound will likely degrade and rot away long before it racemises. But even that, for alanine at room temperature, does not happen on any timescale that you can observe.