As others already suggested, for your needs, stay away from f/4 lenses. Get the fastest you can. This means expensive and heavy but it will pay off.
If you want a single lens for birding and stage work and if budget permits I would get 70-200mm f/2.8 IS II. Sharpest zoom ever and very versatile. It also works very well with extenders.
Another idea: Again as others suggested, 135L f/2 truly shines on FF and would be a great lens for stage work if you don't really need the flexibility of a zoom. You could add a 100-400L (or 70-300L) for birding (without converter) to it, for around the price of a 70-200mm f/2.8 IS II + extender.