I use a 70-200 Is L 2.8 and it is a great bit of kit.
I also use lens (with and without convertors) for larger mammals or habitat/environment photographs.
F2.8 has several advantages most of which have been discussed but. If you think 200 F2.8 has a narrow DOF then try a 500mm F4.0 or 600mm F4.0!
- quick atuo focus (my 5d gives a cross-hair focusing with such a fast lens)
- reduced depth of field is an advantage (used creatively)
- still F4.0 with 1.4x convertor or 5.6 with 2x (autofocus and IS still work)
- you can always stop down - F8.0 for your bird example but a slow lens can't be speeded up!
- add 12mm or 25mm extension tube for great macro lens
Don't give it up just yet!
jason