Starting Young

Don't bother trying to catch the flying ones. Some are on the ground (Shift+V helps to spot them) but unless the area is overly full with other players, simply head to the top of the mountain, there should be several hatchlings on the ground.
Don't shortcut the stepstones, use the last one with the upwind, this will give you a 2 minute buff that a) increases haste by 100% and b) lets you jump as if in an upwind every time you press space - that way you can reach the top of the mountain very quickly.

Also note that since 4.0, you can bind a key for "Interact with Target", which basically simulates a rightclick. (Credit to Talonic, I'm just paraphrasing their comment further down that might not be that noticable anymore in the new comment system)
11.0 edit: by now, there is also a general "Interact" keybind which is even more helpful to interact with small targets.

This. The birds are much easier to obtain at the peak of the mountain. Plus they have a fast respawn rate, so it'll only take you less than a minute to obtain all of them

This is a really good examples of "Quests that become much easier when you bind "interact with target" to a key". Target tiny airborne birdie, air vent jump, then hit your interact key. No mouse click necessary for us slow folk who can't target things to right click.