I dont think it fully works, as depending on what is happening, the whole party could be impacted by the player playing that race, and other players may not be happy about it.
I ran a Dragonlance campaign recently, I said based on the world, there are the common races that are native to the world, but any race available in 5e is playable, they would just be outsiders visiting Krynn, and depending on the race may get quite a bit of attention, possibly unwanted, by npcs around them because of their unusual appearance.
I also said that Dragonborn are a step further, as due to the emergence of Draconians, many may be suspicious of another dragon looking creature, and so would make some interactions harder, and some potential routes / opportunities may get close off as a result , though campaign will still be doable.
If a player did choose to be a Dragonborn, then it would impact on the whole party for being associated with that character. In this instance, it may not have concerned the DM so much, but may have made other players unhappy, and in that instance, who takes precedence?