So if you are willing to work with a Player to fit a Dragonborn into your game then where is the problem?
Why are you self describing as a "lazy" DM when it seems to not be the case?
Your stance seems to be that, if a player wants to play Dragonborn/Monk/gunslinger, they should be allowed, regardless of how that fits the actual world. I can agree, as long as they are willing to work to make it happen. I cannot agree if it is simply them wanting, and not doing anything about it.
If I have misunderstood your stance this entire time, I apologize. Easy to miss subtleties in 200 pages of posts.