Only 3 rounds if you don't land, again meaning you can fly all you want as long as you stop every 3 rounds.... that's an incredible amount of mobility.
1) Rivers - Non-issue
2) Scaling a mountain - Trivial
3) Got a combat with melee only monsters - Fly up for 3 rounds and be completely immune to the combat.
For only one person. The rest of the party is left behind. The only way that the flying dragonborn can help is by hauling up ropes so the rest of the party can climb. And the only way this actually makes rivers and mountains a non-problem is if the entire party has a flying speed. If they don't... well, no party really likes staying still will one or two people do all the scouting ahead. You'll either get the players talking it out or inter-party conflict.
You're also going to end up really ticking off the rest of the party if you avoid all combat by flying above it (and if you give XP for killing monsters, that PC gets no XP). If you are instead using missile ranged from afar--well, that's not much different from being a caster who hangs out in the back away from the melee combatants. I don't know how LU is dealing with firing into melee, but I think the RAW/RAI for 5e is that if you fire someone who's protected by cover, and you miss, you hit the cover. Combat may be enough to count as at least a bit of cover when attacking from outside of it. And I for one would say that a miss when firing into a melee means you hit a
person next to your target--which could be a teammate. If I had to deal with a flying PC, I'd
definitely use that rule.
Also, don't forget that when you're flying, if your speed becomes 0 or you're knocked prone, you fall. If you use your wings to slow or stop your fall (as per Xannies'), that counts as a round of flying, which could mean fatigue. Falling causes damage, possibly enough to trigger Massive Damage (there's a System Shock table in the DMG).
And a bit of meta-gaming stuff: on the ground, you can be surrounded by 8 creatures. In the air, you can also be surrounded by creatures below and above you. Assuming a grid, that's an additional 9 creatures above you and 9 more below you.
Finally... if it's logical for the enemies, don't have them be melee-only.