Why is diatomic oxygen such a powerful oxidising agent if it is stable?

I am very confused - any electrons added to dioxygen must inhabit an anti-bonding molecular orbital which leads to destabilisation of the molecule. So why is dioxygen such a powerful oxidising agent?

I need to understand this to try and figure out why nitrogen monoxide is readily oxidised by dioxygen to nitrogen dioxide. At the moment my reasoning is that the more electronegative oxygen molecule attracts the unpaired electron from the NO molecule, forming NO+ and causing the now ionised dioxygen to separate to O- and an oxygen radical. The NO+ and O- then react to form NO2. This can't possibly be the mechanism through which the bonding occurs, but I can't find any sources online. Can someone please explain this to me?
 
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