I really enjoy using Nikon D500, an APS-C camera, it is my best companion of bird photography.
The endless discussion between Full frame and APS-C sensor makes no sense. What I really concern is the performance in real situation. Camera is a complicated tool and sensor size is only one of the components. In addition the AF system, responsiveness, weight and availability of lens are more important, they are the keys determining whether you can or cannot successfully take a clear picture of bird.
So after using the D500 for about half year, I gradually develop a new habit in taking bird pictures.
1.) Slow shutter and low ISO
We use to set a higher shutter speed to trace bird. In general we will consider shutter speed from 1/800s to 1/1000s is safe for birding. However, in real situation, where light is relatively dim, a safe shutter speed will push the ISO in high level.
The safe shutter speed for Nikon AF-S 300m f4 lens, handheld, should be 1/450s (crop factor of APS-C). However, after repeated testing, without VR, my limit is around 1/200s so that I can save 1 stop of ISO.
2.) Get closer
Well always try to get closer to your subject (nonsense,i know). Bird, especially a smaller one, are very active. A slow shutter speed may not be able to frozen their action. Unless you buy yourself a long tele lens (e.g. 500mm or 600mm), a better way to do is to get closer to your subject.
Using a tripod or monopod is a good idea. In addition, I found the quiet mode (Q mode), which will keep the mirror all the way up during exposure, of new DSLR is quite useful in reducing shutter vibration. Personally I can attain a higher successful rate when using the Qc (Quiet Continuous).