Last Sunday, I returned from a photo trip to Nepal that included lots of street life and lots of wildlife, including plenty of birding. First time for me to take my D500 on a wildlife trip. The experience reaffirmed what I learned over many years of shooting: as Marcel hinted, the lens is far more important than the camera when it comes to birds. If BIF is your thing, consider your lens choice at least as carefully as your choice of a body.
Where I have a different viewpoint is that when it comes to birding, I, like many others, view Nikon's 153-point AF system as superior to everything else out there because of its speed and accuracy, to the point where the only camera bodies I would even consider are those that have it: D500, D850 and D5. The D810 is a great camera, but not my choice for BIF.
This statement applies ONLY to BIF, not to general shooting: there is not much to be gained by using a D850 over the D500 when shooting BIF. You're very likely to shoot in DX mode anyway, where both bodies have about the same resolution and effective sensor area used, and while more real-life tests are still needed for the D850, the specs don't hint at much that would create a notable difference here.
As far as your other point goes: yes, all of the evidence points to the D850 being a great all-around camera that handles almost every shooting situation very well. It combines outstanding AF with high resolution, high dynamic range, and very versatile features overall. I myself am tempted to replace my D7100 with a D850 for that reason - but my longest lens would stay on the D500 since I prefer carrying two bodies.
One last thought: high resolution in a body is great, but for most non-professionals, that is for a different reason than you may think. As I am writing this, I am looking at a shot I printed in 24x36" format years ago. It shows fine resolution and detail but was taken with a 12 megapixel D90. With today's 20-24 megapixel cameras, you're not going to see a print quality difference unless you go to extremely large print sizes, which few of us ever do. What high resolution gives you that often matters more is the ability to crop and still retain a great shot with decent resolution. I know (and respect) some purists who want their pictures to be "just right" in-camera. I'm not one of them. I often end up cropping shots, either because I was too far away to get full-frame, or because I had the wrong lens on the body but the shot couldn't wait, or for a number of other reasons. That's where the D850 really excels: it gives you more flexibility than anything else Nikon offers at this time.