As an aside, there is poor evidence that consumption or topical application of anything artificially high in anti-oxidants does any good for healthy people, although you see "antioxidants" used to sell crap all the time.

Maintaining the right balance between free radicals and anti-oxidants is the sort of thing biology is pretty good at on its own. Trying to put a finger on the scales in one direction or another is probably pointless at best.

When thinking about it it seems pretty weird. Wouldn't most free radicals either enter via lungs or digestive tracts? And those will have plenty of stuff to react before any anti-oxidant can get to work... And anti-oxidants spreading through body on other hand seems bit worrying.

Most free radicals are products of reactions inside the body and the body does have mechanisms to deal with them, even if not 100% sometimes.