Are all organic molecules with odd-numbered electron count radicals?
By IUPAC definition of radical, the organic molecules with odd number of electrons must have at least one of them unpaired and are therefore radicals. The fundamental reason is the definition.

Bear in mind, that other spin states must be taken into account, as simplest example the oxygen. It exists as a triplet, so two unpaired electrons.

The situation gets different, when you do have a metal in your compound, but it is out of the scope of the question.