600 for wildlife with 1.4 or 2.0x. 400 focuses closer and with the F2.8 (vs F4 for the 600) but is shorter.
I suggest you consider 200-400 as a starter super tele. Not quite as fast as the 400 F2.8 but a lot more flexible.
I have and use the 300f/2.8L II, 400f/2.8L II, 600f/4l II & a few months back acquired the 200-400f/4, I shoot wildlife.
The 300f/2.8 I'll keep as it's amazingly sharp, light & flexible with converters, the 600 I'll keep as the 200-400f/4 wont get the range without adding additional converters, and although the Lens produces good IQ with the built in 1.4x + added 1.4x, the 2x converter in this mix is marginal at best. There are shots on my last trip that the 600 + 1.4x just handled so much better than the 200-400f/4.
In this mix I sold off the 400f/2.8L II, it's a lovely Lens, but I found I was always going to the 200-400f/4, it's a one stop loss of course, but I found it wasn't a real issue and easy enough to work around.
I would suggest looking at selling both Lenses, and buy the 200-400f/4 with an additional 1.4x Converter to give you 784 f/8 at the long end, this will not be as sharp as your 600 + 1.4x (840), but you need to pixel peep to see the difference.
For Wildlife, sports, the 200-400f/4 is the complete Tool (I'm imagining the sport as I don't do sport Photography), but for Birds ?? not sure you can beat the 600 other than maybe the 800.