It's been my experience that Canon and Nikon are so different ergonomically that Canon users pick up a Nikon and go "ugh" and Nikon users pick up a Canon and go "meh." This is absolutely something you need to by getting your hands on stuff, testing how it feels, how intuitive it seems to you.
I think the far more crucial issue isn't the body, it's the lens. Depending upon what type of birds you shoot, 400mm may be too skimpy (like Bald Eagles or other raptors). I'd be clear on the glass first, and then see what system offers those options within your price points.
I get your point about DSLR's not being the future of the camera industry. Which is why there are bargains to be had. Frankly, given the issues you have with your D7200 (about focus speed), what I'd recommend is another DSLR: the Nikon D500. Superb focus, good FPS, excellent ISO range, perfect for shooting wildlife. And you can get it on-sale now (new). And if you stay with Nikon, then an FTZ adaptor and those lens will work for the mirrorless you ultimately upgrade to in 2-5 years.
Personally, I think for wildlife photographers, you want two bodies. One with your long glass, another with shorter range or landscape/wide angle capabilities. That's why I suggested the D500--because your D7200 then becomes the second camera with the wide angle lens on it.
And if you think it's a mistake to buy a DSLR now rather than a mirrorless, I think mirrorless is changing so quickly than any serious photographer who buys a mirrorless body that isn't a Z9 or an R3 now will be upgrading in 2-3 years anyway.
BTW, I have a D7100, not a D7100, very different sensors, and I think the ISO sensitivity is worse on mine than the D7200. Also, I don't want to deal with having 2 cameras, I don't care to lug that much gear and weight around, if I need to switch lens, that's fine, all of a matter of good planning and flexibility. It's not like I am a professional looking to sell my wares. I do photography for myself and family.
Look at Fuji. The X-Tx series for a MILC and the X-100x series for a fixed-lens go-anywhere camera. Don't be shy about considering trailing edge cameras--new or used. New stuff rarely delivers better than incremental improvements at considerable expense. Spend the savings on glass--my MO across several systems.
Birds can be SMALL.
Depending on small and how far away, the 400 could still be too short.
There is a guy on the micro 4/3 forum who uses a 100-400 to shoot birds. That would be equivalent to a 200-800 on a FF camera.
Try this trick.
Put the 80-200 on your D7100. Set to 200 and look at a bird about the size you think you will be shooting, at about the distance you think you will be shooting.
Full frame = 300mm on a FF camera.
A quarter of the frame = 600mm on a FF camera.
I think I need to put into some perspective into my use case, as I think we are getting hung up on bird photography here.
Here are my priorities in order:
1. Shooting pictures of people, especially family, indoor, outdoor, difficult lighting situations, and kids.
2. Street Photography.
3. Landscape.
4. Land based wild life.
5. Birds if the opportunity presents itself. This will be more a function of the lens than the body.
Anyways, I must also point out that I have had my hands on a Canon 7D Mark II, and I have to say I liked how it felt in hand, so this is a non-issue from my point of view. However, it would be worth maybe taking a look at. To be honest, it would be something to get used to, adapting to new things is something I'm used to dealing with.
Again, I'm not interested in going with another APS-C system, I really want something that can handle low light situations so the Nikon D500 isn't really a logical choice for me. Sure it could use my current batteries and lens, but I would still be stuck with the slow lens that I have:
NIKKOR 16–85mm f/3.5-5.6G ED VR - This one is slow because of Aperture.
NIKKOR 80-200mm f/2.8D ED - This one is slow because it's a screw drive auto focus setup. Much higher end cameras need to focus this one quickly. Which I do not want to do, because I want to go mirrorless and it wouldn't be supported anyways.