Jon-A:
I was recently in your situation.
After years of being out of serious photography (and those were in the days of FILM), I decided to get back into it and in the summer of 2019, I duplicated my old film kit with digital, and the camera that I bought was a Nikon D5600 and an 18-300mm f/3.5 lens. By the fall I added a Nikkor 200-500mm lens. This past summer, I joined another online wildlife photography group and entered a bird photography contest. I found myself growing increasingly frustrated by the 5 frames per second (fps), and the somewhat slow autofocus. I was missing a lot of shots, particularly of birds in flight (BIF) with the D5600. I was originally considering upgrading to either the Nikon D7200, D7500, D500, or the D850. After getting a lot of input from members of the group, reading a lot of reviews, and watching some of Steve Perry's YouTube videos, I decided on the D500. I've had it about 7 weeks now and I have NOT regretted it!
The D500, being a crop-sensor camera, gives me that 1.5 times (x) factor that gives my 200-500mm lens a 750mm reach, which I find invaluable. Yes, it is only 20.9 MP vs. the 24MP of the D5600, but you're not going to notice that at all.
The fact that I can now get 10 fps has made a HUGE difference in the number of "keeper" shots that I have of birds, especially BIF. You need to have a fairly fast SD card to get 10 fps, but I used the 250MB/s SD cards that I'd been using in my D5600 and was getting that speed (although I did upgrade to 300MB/s SD cards to make sure I was in a "faster zone."). You can also use the even faster XQD card (avg. 400MB/s speed) and/or an SD card at the same time. With the 200-shot buffer in this camera you can shoot like crazy and still be unlikely to have the camera choke. You can also (in the menus) tell the camera to use either card separately or, if you have both an XQD card AND an SD card in at the same time, have the camera use the XQD as the primary, and immediately backup the photos to the SD card. The caveat for me here was that XQD cards are more expensive, AND you'll need to buy an XQD card reader to download your photos to your computer. My iMac has a built-in SDXC card reader, so for now, I'm keeping things simple and just using SD cards.
I've also noticed that I get better low light shots with the D500. I tend to shoot in the late afternoon near sunset when the birds are getting ready to roost for the night and there's a fair amount of bird activity. I was amazed at some of my shots when I was losing the light. Much better illumination than I had expected, and much better than the D5600 had given me!
One drawback has been that the D5600 weighs one pound and the D500 weighs TWO pounds. Couple that with the 5.1 pounds of the 200-500mm lens and that's over 7 pounds of weight around my neck, and it can be a challenge whipping all of that weight around when I'm trying to catch a raptor (or any bird) that just flew in out of nowhere, and get the camera up, sighted on the subject, autofocused (I use back button focusing, by the way), and then get off a couple of bursts of shots! I'm getting better at it, but it is a skill that one must work at.
BOTTOM LINE: I love my D500 and I am VERY glad that I got it.
• It is $1,000 less than the D850 (which was my runner-up),
• the crop factor gives me the reach I need,
• 10 fps means I'll get more keeper shots,
• I have two memory card options, and a 200-shot buffer,
• and it feels more substantial in my hand than the D5600.
I hope this helps.
And before I forget, assuming your shooting technique is good but you still have shots that are somewhat soft or noisy, I would highly recommend Topaz Labs' Topaz DeNoise AI (that's A-eye), and/or Sharpen AI. I think they even offer a free trial. DEFINITELY worth the investment for tweaking the crispness of your shots!