Minecraft is awesome, but I think adding a flying mount would be cool. Balancing my suggestion to coincide with Elytra -- it will not replace the need for one but rather require one to make the most use of this mob. For balancing I can suggest a few options.
limited in terms of speed until an endgame item is found in the end and equipped.
To require the use of elytra to ensure your safety instead of replacing it, it shakes riders off if hit in midair to balance out.
Basically it could have its use as available flighted travel midgame but it isn't as safe or as effective until you already get an elytra due to the rider being able to be shaken off and fall to their death if the mount is hit while flying.
This or it becomes available in a sky type dimension only accessible after the end / requiring an elytra to access anyways.