The class I teach has recently covered oxidation of lipids and I've not found a satisfactory answer to this question. BHT and vitamin E are radical scavengers because the phenolic hydrogen can be abstracted to form a stabilized radical. Why is the phenolic radical more stable than the benzylic radical that could also be formed?
I could not find the one-to-one comparison for which of the phenoxy or benzylic radicals are more stable and by how much. However, for the two examples you gave (BHT and vitamin E), I think there's a good explanation.
For BHT: the neighboring groups are tertiary-butyl groups, so there's no alpha-hydrogen to abstract. Of course, there's a methyl group on the para-position, and there's an old reference that indicates that you can form radicals there for BHT (2,6-Di-tert-butyl-4-methylphenol) on the methyl carbon from a rearrangement of the phenoxy radical. So I wonder if you can also be getting the benzyl radical on the para position. Either way, the reference
(pubs.acs.org/doi/abs/10.1021/ja01612a019) indicates that the benzyl radicals (from multiple BHT parent molecules ) will recombine, i.e. forming a dimer of BHT.
For Vitamin E: it appears the literature has plenty of examples of seeing phenoxy radicals. It makes me wonder if you don't see the benzyl radicals because they are prone to fast rearrangements and dimerization.
Sorry if that's not a complete answer. I did not see any reference that compared one-for-one the phenoxy and primary benzylic radicals (by the way, tertiary benzylic radicals are quite stable. The extreme case being the triphenylmethyl radical (en.wikipedia.org/wiki/Triphenylmethyl_radical).
Phenoxy radical βC- (or βO- ) bonds are weaker than those of the corresponding benzylic radical bonds by ~10 .kcal mol-1. As a result, phenoxy radicals are expected to be far more reactive than comparable benzylic radicals. Breaking of a β-bond in these radicals generates rather reactive unsaturated molecules, so that these bonds may serve as H-transfer catalysts (cf. subclass 1 of benzylic radicals with weakβ-H bonds). Therefore, βC-C and βC-O bonds in phenoxy radicals are susceptible to more rapid bond scission than their corresponding bonds on benzylic radicals. In layman's terms: The phenoxy radical has weaker bonds overall than the benzylic radical making it more reactive (easier to break up bonds) and therefore likely to be form and be part of a reaction before a less reactive benzylic radical. (Source: Chemistry of Conversion by Richard H. Schlosberg.)