I've got an Olympus em5 iii with the 75-300. It's not the fastest lens, but if you're in bright light it work well for everything I've done with it, and the whole set up is smaller and lighter than my old Fuji with the 55-200. Handheld image stabilisation on the Olympus cameras is also impressive even at these focal lengths
I was also going to suggest m43, although I understand that the contrast-detect auto-focus (CDAF) used on most models is not the best for bird-in-flight photography. At least that's what I recall from reading the dpreview forums. OP, it may be worth a try after you do some more research. Don't limit yourself to Olympus only though. Lumix (Panasonic) also make excellent m43 cameras and lenses.
The body I have uses a hybrid focusing system. From what I understand it uses PDAF for finding initial focus then refines the focus point using contrast. Not all bodies do this though. The "pro capture" mode I've found incredibly useful. Basically buffers in busy mode, until you full press the shutter. This allows you to frame a shot and get focus where you want it, but only capture the frames immediately either side of the shutter press. Other cameras may do this, but it's new to me and very fun
Right, as far as I'm aware, only like a couple of m43 bodies (Olympus only, I think?) have PDAF, although I haven't really kept up with m43 releases the last few years. It's been long-requested, but Panasonic, at least, seemed to be unwilling to do it, claiming the potential for image artifacts caused by the PDAF pixels. That's a real possibility, but they finally relented on their latest L-mount body (S5x or something?) after the previous L-mount bodies had that the auto-focus in particular being widely criticized. Perhaps the replacements to the G9 or to the GH6 will have PDAF.
One possibility would be getting a used APS-C DSLR from Canon or Nikon, both come with a wide range of their own and third party lenses. Since both are wide spread they're comparatively cheap second-hand. That might vary depending on where you live though.
I'm not sure about the current state of things, but until a couple of years ago DSLRs had still an edge over mirrorless regarding wildlife and sports.
The current generation flagships and top prosumer mirrorless have action AF probably close enough to on par vs the previous flagship DSLRs. However, anything below that level may still be pretty slow for fast action.
There's no replacement for reach, but if you really want to fill your sensor you're probably going to always be wanting for more unless you go after larger birds. One of my bodies is a Nikon J5 (1" sensor/2.7x crop factor). I have the FT-1 adapter and with my 70-300 get a FF EQ 189 - 810 and I find myself wanting way more. I don't have the time to camp out and wait for birds to me due to having younger kids. At 30 feet my sensor is fairly well filled if I'm aiming at a finch, but the only way I'm getting that close is if I'm using my bird feed as bait.
I'm not sure if this is viable or too expensive, but we do like Sigma and use their lenses professionally. This has just been announced: https://camerajabber.com/sigma-launches-100-400mm-f5-6-4-dg-dn-os-c-for-fujifilm-x-mount/
It might be a bit slow at the long end though for birding, but I'm sure it'll be razor sharp.
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