I'd go broad reading.
First, because I feel it's correct as rules-as-intended. I think it's just located there because that's where it first comes up when creating a character, instead of burying it elsewhere away from it's most likely use.
Second, because characters can gain still from classes beyond 1st level and shouldn't be penalized for having a coherent concept where a needed background skill is delayed until whenever because that's when the class would grant it.
Third, the only coherent reason not to I've seen is so players don't optimize to pick off their list, but customizing a background and picking ANY skills/tools/lanaguage and features (or working with DM to make new features) is already a basic rule (and not an optional one). PHB, pg 124.