Just for the record, it doesnt fits your needs/demands, but the actual best all-around camera would IMHO be the Nikon D750, right now. You dont actually want the most general camera, though, anyway, which your first posting already reveals quite quickly. Cameras without interchangeable lenses will never be "all-around".
I would also like manual controls so I can learn real photography [...]
Forget that requirement. Its not realistic. Cameras with good manual controls are pricey. They also have interchangeable lenses, or are compacts for photographers, with a high quality prime lenses - not what you want.
The Sony RX100 III seems to be mostly what I'm looking for though more manual control buttons and better zoom would be nice.
If you can stomach the size and price, Sony RX10 or that new Fuji, wasnt it called FZ1000 ? They offer more zoom range, pretty good image quality, and more tolerable manual controls.
So is mirrorless the way to go now? It seems like they are a good balance between everything.
No, they sacrifice AF performance (at least when we talk about moving subjects, especially in low light) and an OVF for more compactness and simplicity of the camera construction (which theoretically saves cost and yields higher reliability, though in practice until now the much smaller series still kills the cost advantage). For the most general camera, this is not the way to go just yet; technology will have to improve more before it will be.
I think the Sony RX100M3 is a pretty good all around performer. I've been wanting to pick one up for when I really need to travel light or don't want to carry my DSLR with me.
I actually would prefer the RX100M2. The buitlin EVF of the M3 is IMHO near unuseable (stalls when focusing) and the focal length too limited. The M2 has the same sensor, and in low light I can limit myself to an 28mm equivalent just fine. How well the EVF of the M2 works I havent tested, but its separate and optional anyway. Mind in bright light you might have a hard time to see the back monitor.
Agree and disagree, its awesome with its FF sensor but since it has a 35mm fixed lens then its more directed at street photography. So it depends of your style of shooting. Birding, portraits and sports this is just not right camera. Landscape and city street its perfect.
While the most classic focal length for portrait is probably 85mm, you can shoot portraits just fine with a 35mm, it will just be environment portraits, i.e. you'll see a lot of the location you're shooting the person. Meaningful portrait focal lengths really go from 20mm to 400mm, possibly even longer, always depending upon what you want to do.