My experience with playing rare stuff bears no relationship with that, and it's not because my group and I "don't care".
Thing is, non mundane stuff isn't rare, when taken as a whole, in the Realms. Any given individual part may be, but as a whole, not so much.
Look at it this way. Rather than looking at a tall white guy in a fairly monolithic Asian country, the dragonborn is like a Sikh in the full traditional garb, dagger and bracelets and turban and rad beard and all, wandering into a bar in the US, in a town with no Sikh population, but in general a diverse demographical makeup. Does the Sikh turn some heads? Sure, most people in the bar have never met one. Depending on the bar, someone might make a rude and ignorant comment about people from the Middle East. If a backward enough place, people might even generally mistake the man for a Muslim, and due to rabid mouth frothing racist idiocy, treat him poorly.
But, the reactions are not going to be the same as they would if the same man walked into an establishment in boot scoot Nebraska, circa 1950. (and now I have "Boot Scootin Boogie" stuck in my head...I hate that song)
FR has been for some time a place where there are plenty of non human, even non Tolkien, people running around in enough places, killing monsters and running bakeries and what have you, that while a person of an unknown race, obviously from a far off land will attract notice, it's nothing on the order of a teifling walking into the Prancing Pony.
In other words, there are plenty of ways to deal with players playing things which are rare in the game world, without it going the way you've described.
edit: and without losing the "stranger in a strange land" vibe.
But also, if a group doens't want to deal with that, and just wants to assume that dragonborn are well known enough that people don't really freak out or even stare much, and just get's treated like a foreigner, at worst, that's fine. The point of playing a dragonborn isn't to play a stranger in a strange land for everyone. For many people, it's about things entirely unrelated to that, in any way.