Why do Grignard reagents act as sources of nucleophilic R− and not Br−?
In all likelihood the halide ion does attack the substrate.  But, the product such a reaction mode would form does not accumulate whereas the alkylated product does.
Recall that in a nucleophilic reaction halide ions are good leaving groups, at least if you stay away from fluorides which are rarely used for Grignard reagents.  So while the halide ion may add to the substrate, it may also be displaced if an alkyl-anion moiety from another Grignard molecule comes in, or possibly the halide ion may just launch on its own to attach to a magnesium atom (which, in a Grignard reagent, acts as an electrophilic center, for example by adding to the oxygen end of a carbonyl group).
In contrast, when the alkyl-anion moiety attacks, the carbon-carbon bond it forms is strongly favored thermodynamically and a lot harder to break; here we do not have a good leaving group.  So the alkylated product rather than the halogenated product is what accumulates.