I fell ass-backwards into Micro Four Thirds and can't be bothered to leave.

Right now my main shooter is an Olympus OM-D EM1X. You can pick these up for about $700-800 even though they were the top-end pro model only four years ago. At times I also use an Olympus PEN E-PL1, which is an ancient early MFT mount camera that will produce some damn fine images on an extremely low budget ($50-60 body only, maybe $100-120 with a kit lens).

As others have said, the lens is really everything. I threw down on the Panasonic Leica 100-400mm and have not regretted a single dime of the $800 or so I spent for it on eBay. I've seen them go for less but mine was in very good shape.

There are tradeoffs with the MFT system. First, it's a smaller sensor. It will be claimed the low light performance is poor, but this is again down to the lenses you have available and the native ISO of your sensor. The other tradeoff is kind of a positive, kind of not. It's a crop sensor system, meaning that it doesn't "see" all of the light entering the lens. It's a smaller image relative to a "full frame" camera. On the one hand you're getting a "smaller" image but on the other it basically doubles the focal length.

So that 100-400mm lens I mentioned gets me the equivalent of 200-800mm of effective zoom in a package that's smaller and lighter than my sister's Canon setup with even a sub-100mm zoom.

Another cool benefit of MFT is that it's dead easy to adapt basically anything to it. As long as the lens is manual there's probably an adapter for it, which massively expands the types of lenses available.

Outside of that it's all upsides. I mean... you DO have to deal with the constant "Micro Four Thirds is dying!" and "LOLTINYSENSOR" garbage from those of a far less refined nature. But they can piss up a rope. lol

Wildlife is going to be pretty lens heavy. You'll want at least 400mm equivalent.

As r/photography likes to say, the equipment is really personal to you. For example, I absolutely hate using Canon cameras, something about their UI just puts me off; I'm a good deal slower when using them. If possible you should try out some cameras to see what sticks. If you're serious I'd suggest renting something that catches your eye and is around your budget to try.

If you're on a budget a DSLR is actually a great pick; you probably can get a higher level body used with some really cheap 80-400/100-400/150-600mm type lenses now. Saw on my local Amazon a Tamron 150-600mm G1 new being cleared for like under US1k.

Edit: Also, stating your budget will really get you more targeted recommendations.