I don't know just how swift the AF is on the latest iterations of the D 3xoo family, but if you're looking for something in DX, you might consider a D7100 or 7200. I have a D7100 and the FX 70-300 AFP, and my wife has the same lens on a D7200. We've both had pretty good luck with birds in flight using this combo. The AF is a little easier to manipulate than that on the lower end, and hers is a little faster and better than mine. My only real-life comparison here is from my previous D3200 to the D7100, and though I got fairly good at panning and lucky with the occasional bird using the older rig, there's a definite improvement there. The D7200, now discontinued, is a nice machine with lots of capabilities and controls that the lower ends lack, and the sharp 24 megapixel sensor has a lot of cropping room. Up until the D500 came out some critics counted this as one of the best wildlife cameras out there. Couple that with the 200-500 / 5.6 and it's pretty hard to beat. Whether this is now met or exceeded by the latest D3500, I can't say, but the D7x00 does have a control advantage.
I have both the 70-300 AFP-FX and the 200-500, and both are nice to use, but the 70-300, because it is lightning-fast focusing, as well as easier handling, tends to get better results within its range for things like birds in flight - at least easier.
Way back, using a famously needle-sharp old 400/5.6 AI lens and a matching Nikon 1.4 teleconverter, I did a little study to try to determine the sharpness difference between TC and crop on the D3200. Of course if you have to crop way way down, the TC wins because of pixellation, but if you compare the lens/TC combination with cropping the lens to match the TC, it's pretty much a dead heat, with the crop edging slightly ahead if you crop further or need the speed lost by the TC. Not to say you shouldn't ever get a TC, but before springing for it, I'd check how far you can get away with cropping.
As an old film and slide shooter, I used not only to be unable to crop but to disparage it, and it's taken some time in the digital world to shed that prejudice. Now, with a dense DX sensor, and often the hurried decisions one makes when traveling, I am used to it, and often shoot a little "loose" because I'd rather crop something out than regret that I didn't get it all in.