A Tiefling variant (SCAG) gains flight at 1st level. Is this game breaking? What kind of problems could this produce? I know that Fly is a 3rd level spell usually granted at 5th level, and draconic-bloodline sorcerers get wings and flight at 14th level.
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While it may not be "game breaking", it is unbalancing enough that it is disallowed for Adventurers League play (same as the Aarakocra PC race, which also grants flight):
As you noted in your question, flight is an ability that usually only comes available at higher levels, and usually for a limited duration. Being able to fly enables you to bypass all sorts of challenges that ground-bound PCs face.
Here is what flight does: it makes the player character immune to certain types of monsters. For example if the party gets attacked by bears, or wolves, or lions, or tyrannosaurs, your tiefling variant can fly into the air and now the monsters can't hurt him.
Usually this is not a huge problem, because the monsters can still hurt the rest of the party, so you still have an interesting combat. But what if two-thirds of your party have flying? What if the party gets attacked by bears, and most of the characters say: "nope!" and fly up into the air, and then the last remaining non-flyer gets attacked by all the bears at once? (And then the flyers kill the bears by shooting them from the air?)
What if your tiefling says: "hey guys, I think there are bears over there, let me scout ahead" and then he goes and fights the bears by himself, shooting them from the air until they die or run away?
Flight might seem like a big help in solving certain logistical challenges such as crossing chasms: the party can have the flyer secure a rope to the other side, then rope-climb across. On the other hand, if they hadn't had a flyer, they could have just used a grappling hook to secure the rope. It's not clear that the flyer adds much to the equation.
Flight does let the character avoid falling damage, but characters very seldom take fatal falling damage anyway, so this doesn't seem game-breaking.
Flight does allow the party to perform aerial scouting: while outdoors, they'll be able to spot most enemies from a distance, and they're much less likely to get lost. Of course a wizard's bird familiar could do the same thing. These abilities don't seem game-breaking either.
Overall, if you know what encounters the party is going to be facing, and it's all flyers or things with ranged attacks or battles in a room with a low ceiling, then you should be fine.
So flying is definitely powerful enough to be disallowed for Adventurer's League play (as pointed out). Adventurer's League is quite focused on lower level play though, as evidenced by the number of adventures targeted at levels 1 to 4. So I think this is really specific to both the adventure and the implementation.
At early levels, flying can outright break some encounters, especially if the villains were not equipped with ranged weapons. That stated, if you're running around a dungeon, flying can be very limited with 10 foot ceilings and no place to spread your wings.
You mention the Fly spell becomes accessible at level 5. But Druids at level 8 can also fly. In fact, by level 8, Moon Druids can become giant eagles for 8 hours / day with a fly speed of 80. That's well before the Draconic Bloodline wings kick in at level 14.
But please note that Aarakocra and Tiefling wings are not quite the same thing as the Fly spell. Aarakocra flight is quite limited, they can't fly with even Medium armor. You're talking about a 20lbs weight vest completely preventing them from flying. So if you let them fly, but don't let them carry heavy stuff or other PCs, they don't necessarily break all of your puzzles.
You can also enforce that most winged flyers can't "hover". Ensure that they are moving every round or force them to land. You can also rationally enforce a minimum height limit. A bird that occupies a 1x1x1 foot cube typically needs than just that space to take off, likewise an Araakocra probably needs at least 10 feet of vertical clearance to actually start flying.
Also note that flyers can be particularly susceptible to some tactics. A successful Hold Person on a winged flyer will drop them to the ground. Anything that makes them prone will knock them down. As the characters level up, the tools available start to erode the advantages of the flyers. By level 5, every good enemy should be able to deal with flying lest they get destroyed by the Wizard. So they'll be able to deal with your Tieflings and Araakocra.
So I wouldn't call the whole thing "game breaking", especially at higher levels. But flying at early levels can be a little "game-warping" and should be dealt with appropriately. Make sure your goblins and kobolds have slings and bows.