Sulphur Nimbus: Hel's Elixir, a $6 (currently) game on itch.io. It started from the idea of an MLP fangame, but early in development evolved into an original setting.

This is a 3D physics platformer adventure with an unhindered flying character. Your hippogriff, Sulphur Nimbus, is an aerial photographer aboard a cargo ship, which is passing a mysterious atoll on the way to their destination. The crew want you to get pictures of the island, which has a castle that's been abandoned for decades. Unfortunately, after flying over there, a nasty storm builds up and you get zapped by lightning. After a flashback tutorial on how to fly, you wake up on the island shores, your wing is injured, and you have to run to safety, finding out this place is dangerous... so dangerous a resident dogicorn (like a hippogriff, but it's half dog and half unicorn instead of half bird half pony) has to rescue you when a lovecraftian horror tries to take you down into presumably Hel. Waking up in a castle room, your wing is healed, and you can fly again.

Now the game begins. Clear the boss monsters and rout them out of this island, area by area. Break the curse that binds you to this island. Find out what happened here.

What's unique about Sulphur Nimbus is the movement. Running, fighting, and jumping has physics to it, allowing for some parkour stuff to be possible, like running up steep inclines and wall jumping. Flight is realistic. There are no arbitrary limitations, other than a regenerating "flap" stamina. If you can get enough speed to take off, and if there's enough room to maneuver, you can fly. Level designs include lots of caves and enclosed spaces, but also lots of open areas, so being able to fly is a requirement to get through it, while also a challenge. While the game is designed for kb and mouse controls, honestly, a gamepad works very well with this game and is preferred. It also is cross platform, as it is made in java, and includes Windows, Mac, and Linux. The source code is on sourceforge and allows you to build the whole game yourself if you are so inclined.

There's no other platforming adventure game that attempts this, and I have tried every "become birb" game out there. They all are either bird simulators or use flight as a fast travel, but not as a core gameplay mechanic like this.