Thanks to all! Now it is clear why I suck so badly at BIF. Clearly my next step is a 7D II. After that, a fast lens (as clearly advised by jrista and strongly implied by AprilForever). And practice. Thanks again.
BIF is not easy. Not by a long shot. I've been doing bird photography for a few years now, and BIF is definitely my weak spot. I've kind of stopped practicing, as I just don't have the lenses for it. A FAST lens is a big plus, so I really think the 70-200 f/2.8 is going to be a good lens to have, with or without a 1.4x TC. If you have the money, the 300 f/2.8 is great. I am not sure about the 500/4 on a 7D II (that's an effective ~810mm focal length...that's really long, makes for a pretty narrow FoV...I have a hard enough time with BIF at 600mm on the FF, let alone 800mm+). I think a 500/4 on a FF would be ok...it still seems a little long to me, especially if you do not already have good BIF skills (if you had good BIF skills, I think it would be excellent, you'll get a lot more detail...but learning on it would be tough, just keeping the bird in the frame would be tough with a 7D II.)
The 70-200 f2.8 is a lousy lens for BIF. It is way too short. Same with the bare 300mm f2.8. The 300mm f2.8 with the 1.4x will work but it is starting to get heavy.
The premiere lens for learning BIF is the 400 mm f5.6L. It is has the needed reach, especially on a crop camera, is fast focusing and is light weight. Once you have practiced you can move up to the heavier 500mm f4L or the 600mm f4L that you see the serious bird photographers shooting handheld.
Funny thing, I've never heard a bird photographer claim that a lens was too long, too heavy or to long a MFD, yes, but otherwise the longer the better.