Free radicals vs Hund's rule contradiction?
The first statement in the question is not correct. Formation of electron pairs is not favorable. That would lead to putting two electrons in the same orbital and there is some effect from charge-charge repulsion.

Electrons end up paired in bonding or stabilized orbitals because there is space for a second electron, and the overall stabilization that would result from putting a second electron in is greater than the destabilization of getting that electron from somewhere else.