Given that electron affinity is positive, why don't these atoms attract free electrons and "prefer" to exist as anions? Never made sense to me — I could be missing something obvious.
Yes, the electron affinity for group 17 elements (group VII by older IUPAC nomenclature) is generally exothermic, so the uptake of an electron should be favourable.
But as Ivan mentioned in a comment, there is no such thing as a free or lonesome electron. Each electron must first be displaced from another atom — ionised; the corresponding thermodynamic property is the ionisation enthalpy. For every element, the ionisation enthalpy is positive, thus it is always unfavourable to displace an electron from a neutral atom. No exceptions.