The QM language itself is ambiguous (it appears to me), and it carries an aura of mysticism. I admit that it is not easy to rule out the claims, the ambiguity makes it all the more difficult. We can only wait and see whether everything could be explained on the basis of QM.

Electron can cross solid barriers, because the atoms are nearly empty. Solid conductors and semiconductors allow certain pathways. In perfect insulators, when the thickness is very small, some electrons pass through.