The thing to know about the 6D, is it's not really considered a 'professional' camera as such. Sure, it's a fine camera, but features and build quality is generally not up to 'pro'. Heck, Canon only calls the 1D-series as 'pro' cameras, everything else isn't.
When you say 1 stop better, are you just talking about sensor noise performance or AF sensor performance? Even though the center point AF sensor works fine with f/4 lenses, I believe it also has a double cross-point sensors there which are likely only available on an f/2.8 lens. Thus, you still get the point where your AF will likely be improved by having an f/2.8 lens, even if you take the shot at f/4 or f/5.6, even in available light.
That may be true, but I will note - for whatever it's worth - that my 70-300L, at 300mm (where it's quite far from 2.8!), focuses flawlessly on my 6D (center point AF) in *very* low light (including in a situation where a Nikon D600 I was trying simply wouldn't focus at all even with a f/1.8 lens attached). Whether less well-designed/made slow(ish) lenses could do as well I can't say....
True. I won't argue the focus speed on f/4 lenses is just as good in most situations. But for low light action, with an f/4 lens, you're in a world of trouble with ISO values. A whole stop of light could mean ISO 10,000 or ISO 5000, which is a huge, huge difference in IQ. If you can slow your shutter, great, it'll work fine. But not at 1/500.