Thanks everyone for your continued advice! My hands have a history of being slightly shaky, so for sure I will need IS/OS/VC, so the 400mm f5.6L, old Tamron 70-300 and 200-500 are out for me. I will look into the Sigma 150-500 or Tamron 150-600, they seem to create a smooth eventual upgrade path for me such that after enough saving and upgrading, I'll end up with a final 3 lens set of 15-85/17-55 + 70-200 f4L + 150-500/600 + 100mm macro - not much overlap.
I'll look into renting these lenses to try out
One thing to note about IS/OS/VC/VR. More than often, when you are shooting fast birds, you need very fast shutter speed to freeze action. That shutter speed usually will be enough to counter handshake. The problem with handholding for long tele lenses usually is not to combat blur pictures due to handshake, but minute handshake will give you problems in tracking moving birds and keeping them in frame, especially birds in flight. In the end, having a good support solution is a lot more important.