As you said before the problem is that you can't avoid the obstacles because the trajectory of the "bird" depends on the speed of the jump. Maybe you can add two different actions:
- A soft touch to jump a little in the air. With the bird example move the wings a little but not fly.
- Continuous touch to go down as in the original "Tiny wings".
With those actions you can place different kinds of obstacles (ie. trees and holes to jump, and planes or balloons to go down) and generate the level with a procedural approach. But it will be hard to find the right adjustments.