Oxidation involves the loss of electrons or gain of oxygen. In the case of chlorite, the chlorite ion gains 4 electrons to become chloride and the oxide ions wander off to form water or carbon dioxide, there is no useful oxygen available, so it is oxidative but it's not an oxygen therapy
Spent oxygen is breathed out as carbon dioxide(CO2), and in chlorine dioxide(ClO2) or chlorite (ClO2-) it is also already spent ie in a -2 oxidation state.
It is infact the chlorine that does the oxidizing as it starts off in a +4 oxidation state as chlorine dioxide and ends up as chloride which is -1, which makes ClO2 a powerful oxidizer(it has a high capacity to oxidize) accepting 5 electrons, whereas the oxygen in ClO2 is in a -2 oxidation state the same as in water or carbon dioxide and therefore can't do any oxidizing.
This may explain it, notice at the bottom it explains how the oxide ions are spectator ions, ie they don't change their oxidation state