So far from what I've seen it doesn't.
But I've been building a couple of monks and rangers and its seems like maybe it should.
First, I noticed monks start out with unarmored defense proficiency as experts but trained in unarmed attacks. (pg. 97)
But in the "unarmed attacks" section on 178 it mentions "almost all characters start out trained in unarmed attacks"
With the recent errata that "All classes are trained in unarmored defense" it makes sense that a class that is trained to be unarmored (like the monk) would be experts in that sense, and they are.
However this doesn't seem to transfer to the unarmed attacks, "almost" all classes are trained in unarmed attacks and the unarmed monk is still only "trained" and not experts at unarmed combat.
This led me to also wonder about a elven ranger using bows, should (or would) an elven ranger who took "weapon familiarity (elf)" become experts at using bows? The feat provides training in bows and the class gives training in all martial weapons (ie bows)