Goblins got Fey Ancestry as a racial ability in a very recent book called Monsters of the Multiverse. This game was already in production for over a year when it came out.
The racial abilities BG3 is using seem to mostly come from the Player's Handbook and Volo's Guide to Monsters. Larian likely didn't know that Wizards of the Coast were planning to throw Volo's Guide into the trash and replace it with an entirely new book with new racials.
As for playable goblins.. Of course it'd work. They're playable in 5e and have been playable since 1989, way back in First Edition. They player goblin would just be from a different tribe not connected to The Absolute. If we can have playable Drow, who are also connected to The Absolute and are traditionally evil, I don't see any good reason to entirely exclude goblins from a purely narrative point of view.
