I was about to buy a used 5D3, ended up with a new 6D2 for almost the same money. There are certainly some areas where a 5D3 is better than a 6D2, which for me is primarily ergonomics (I'm used to 5D/7D controls, but that's easy to check in-store to see for yourself), however I don't regret my decision one bit.
The 5D4 is a lot of camera, but you pay for it. The 5D3/6D2 price difference would net you some nice glass, and looking at your glass line-up and what you shoot:
- You need wide-standard covered. You mention the 24-70 or 24-105 F/4 IS. The 24-105 can be found super cheap, but the F/4 zooms don't really interest me personally. Tamron 24-70 F/2.8 VC?
- FF without a relatively fast, wide-ish prime just feels like a waste IMO. I use my 35 F/2 IS a LOT - on FF it's just an amazing beast in low-light. The fast 24mm primes appeal to me too, but my wallet say no at the moment.
- If you're even somewhat serious about the animals/wildlife side of things, that 70-300 IS is almost certainly holding you back, especially with FF. 100-400 or 400 prime IMO. FYI a Kenko DGX 1.4x TC behind the 400 prime on the 6D2 and usable AF across all of the points is really surprising, and I expect the 100-400 would be similar.
For astro, do you use an external display for liveview? If you don't have an LCD for that, the articulating screen on the 6D2 is very handy. It's touchscreen with fantastic liveview AF is mind-blowingly good too and shouldn't be underestimated.
All of these options are huge upgrades over the gear you currently have. Even the original 7D is a big upgrade over a 400D, and if you were to stick with APS-C on a tight budget would probably not be a silly idea in itself given how cheap they're going for.