TL;DR = Just realized everything I say below is "I hate ASI being in Background."
I feel like Class+Race is the core of your character; Background should be an extra layer of bonus goodness, not a key part of your optimization. For example, in the old MechWarrior RPG, you could purchase the trait "Natural", which super-enhanced one of your skills. I think like half-cost advancement, and some "be creative" options. BUT you couldn't pick the skill - otherwise everyone would just be a Natural Gunner or a Natural Pilot. So you build your character, and if you chose "Natural", then you roll to see what you were a natural at - might be Gunnery, might be Tech, might be Stealth, might be Politics!
In D&D, then, if you want to be a rogue or monk or fighter, you build that character. If you choose your narrative, then layer on a Background that matches; or take a random narrative. And the effects of the narrative should be "flavor enhancing", not mandatory. Being a Sailor Monk or a Sage Rogue or a Soldier Wizard should be interesting and flavorful, and in no way gimped compared to being a Thug/Pugilist Monk, Criminal Rogue, or Sage Wizard.
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And, completely against what I just said, my other thought was that when backgrounds and classes overlap, there could be a minor additional bonus, like +1. A Sage Rogue has more skills at his disposal, but the Criminal Rogue is just a touch better at Sleight of Hand because of the double-whammy at character creation. Does +1 on a skill invalidate the benefit of proficiency in an extra skill? Depends on the campaign, but I'll go out on a limb and say "no". I'd rather be proficient (+2 to +6 bonus) in more skills, than give that up for a single +1 to my main skill.