The Nikon D-850 has approximately the same resolution in DX mode as the D-500. The big edge is that you have a choice of FX or DX. Since you have a larger angle of view in FX mode it is somewhat easier to get the subject in the frame than with the D-500 DX. Also, sometimes 500mm on a crop camera may be too long as it is a 750MM FX equivalent so a subject may be too close in DX mode but not in FX mode.
The flip side is that the DX camera is lighter, cheaper and takes 200 shots without stopping at 10 fps, then takes 200 more if you ease off the shutter even for an instant because the 200 shot limit is a software limitation. The buffer never fills up with an appropriate card installed.
Auto focus is interesting. Most say that the D-850 focuses very slightly faster and does low light a bit better BUT the focus points on the D-500 essentially cover the sensor while they don't on the D-850.
So what about in the field?
Getting a bird in flight in the frame takes practice. While a larger angle of view means that you can find the subject faster you still need to get the subject in the GRP autofocus zone. More significantly, a great shot really needs a SP autofocus on the eye. Because focus points don't fill the frame with a D-850, most photographers use BBAF (Back Button Auto Focus) so they can focus, then recompose. My experience indicates no such need with a D-500 DX as you can move the focus point anywhere on the screen. The result is more practice is required with the D-850 to switch AF modes because there is one more button to push with one more finger. For me the D-500 is a two-finger implementation. One figure on the shutter and the ring finger on the Fn1 button. My thumb is not involved. When the bird takes off, I simply release the Fn1 button and switch from SP to GRP. BBAF is really popular but mirrorless photographers are finding it unnecessary because the focus points cover the screen. The D-500 has the same feature. The result is a tradeoff between a larger angle of view but on a slightly heavier camera, shooting less frames per second with greater dexterity required.
To me, the big advantage is that the D-850 can take closer closer subjects. Using a 500pf, the D-850 has a 500mm and a 750mm equivalent while the D-500 only has the 750mm equivalent. I looked at switching to D-850's for this reason but decided on another solution which brings up another comparison.
Comparing a D850 with a 500PF in FX mode with a D-500 Dx with a 300pf. The D850/500pf has a lot more megapixels (45 vs 21) but the D-500 is a pound lighter and much easier to handle. In fact, catching those flitting birds relatively close in even the GRP AF area is worlds easier with the D-500/300pf. My solution was two camera bodies, one with a 300PF, one with a 500pf. This solution was considerably more costly than either single camera setup but allows me roughly equivalent reach and angle of view of a D-850/500pf at F/4 as opposed to F/5.6. The weight to carry is higher but the weight in the hand is not.
So yes, the D-850 is better in that it is two cameras in one, but it is not better than two cameras that cover the same 500mm, 750mm (equivalent) range most useful for bird photography.