There are two aspects of focus.
The first is easy - focus accuracy. When you do focus on the subject, is it consistent and sharp. Here mirrorless has a significant edge and is much more accurate. For perched subjects, static subjects, or slow movement, mirrorless has a big advantage. Even when there are misses, it's usually a small miss and often related to technique. Note - this does not mean every image is sharp as there are a lot of variables other than the camera's focus. Even focus working perfectly has a distribution of errors but that distribution is much narrower with mirrorless than a DSLR. Generally there is no need for AF fine tuning with mirrorless.
The other aspect is focus acquisition. This usually includes a combination of focus speed and focus on the intended target. DSLR cameras use Phase Detect AF and a separate focus sensor. Mirrorless cameras have focus on the sensor based mainly on contrast and in some cases subject detection. DSLR cameras are usually a little faster to focus because they have a dedicated focus sensor. Your D500 also has a separate processor for focus, but the D5 with a very similar sensor and processor focuses faster - largely because it provides more power to the processor. With the most advanced mirrorless cameras (Z9, A1, R3) - you have virtually concurrent focus and readout from the sensor for the viewfinder or LCD. With other cameras it's going to vary and there is more of a lag. Confusing things more, the viewfinder and related AF display lags image focus slightly, so the camera can focus faster than it can also display in the EVF. On top of that, there are a number of technique issues to help the camera focus quickly. If you ignore appropriate settings and techniques, focus may be slower or fail more often. These cameras are not "point and shoot" devices - they may work 90% of the time, but if you encounter the other 10% you need to quickly have a plan.
More recently, cameras have been advancing subject recognition. The flagship cameras are more advanced than other models, but the most advanced cameras are often high end professional television or video cameras. For example, TV cameras can follow a player based on the number on the jersey while photo cameras follow based on subject detection, color, and contrast. Nikon has a series of subject recognition priorities - body, head, then the eye. Eye only matters after the camera has identified and acquired a face. With wildlife, the camera recognizes that it is an animal, what the animal is, then the body, head and eye. For eye detect, results are much better with good lighting such as a catchlight in the eye, but if the subject is large in the frame, it can identify the eye in a profile view. The same is true for tracking - tracking a small subject with a busy background may be very difficult, but a large subject filling the frame with a clean background becomes very easy. My experience is the camera does a great job if the subject is 1/3 of the frame, but much worse if the subject is small in the frame.
My experience is even with the Z6 or Z7ii, I'm able to focus on songbirds in a wooded setting, shorebirds, wading birds, and any kind of perched or slowly moving bird. Flight depends on the size, predictability, speed, lighting, and background of the subject. I also find that my success rate improves a lot if I am very practiced in tracking birds in flight, and much worse without practice.
You should be able to generally keep a Dynamic or Single AF point box on the subject while it's flying, and if you can the camera can invoke advanced capabilities (this is even if you are using a different mode - just an example of the degree of proficiency good photographers have for flight). If you struggle to keep a bird in the frame, the camera is not a magic answer and you'll need to work on technique even if you have a great camera.
For what I shoot, the mirrorless cameras are much more effective than the D500 or D850. Most of my subjects are not fast moving small birds in flight. If that is what you are photographing, the flagship cameras are the most advanced, but enthusiast level models are either available now or will be available very shortly. As a rule of thumb, new cameras almost always have technology upgrades.
