The ability of AF to deliver accurate focusing v MF is interesting. When sports photographers used analogue cameras, such as the F5, my impression was that nearly all the pics they took were in focus. If it was less than 50% then the photographers were wasting 50% of their films and presumably the hight burst rate of shooting that the F5 was capable of would not have helped.
So it seems that out of focus sports shots in newspaper and magazines should have been much more common than they seemed to be. I have never seen any articles saying that the likes of the F5 had only at best a 50% reliablilty and that a lot of film was wasted but that whereas the digital AFs were no better, the ability to shoot in a virtual unlimited fashion with a high MB card now saved them from having to settle for an out of focus picture
Anecdotally and turning now to my own system( Pentax) I rely on the green signal that appears in the viewfinder to tell me that I have my target in focus. I have frequently tried the same AF lens in the MF setting first then thrown the switch to AF and in every case the AF focusing was accurate.
What I have found is that occasionally I have allowed the AF system to focus on the wrong part of the scenes so what I wanted to be in sharp focus wasn't but this was a combination of a system that had a very limited range of focus points compared to the likes of the much more sophisticated F5 and my not paying enough attention to what the focus point had settled on.
Pretty sure your Nikon doesn't do that.
While I can't speak for the Sony cameras, I do (also) extensively use a M4/3, with the primes I've had for years/decades.
First, the existing M4/3 cameras use a different and more accurate (albeit somewhat slower) AF method than the vast majority of (D)SLRs. In fact, in the test I previously mentioned, a M4/3 camera had the most accurate focusing results of all the cameras tested.
That said, manual focusing on my M4/3 viewfinder (not rear LCD) doesn't even come close to manual focusing with a *good* optical viewfinder/screen combination. Better than expected, yes, but the majority of my film cameras will reliably focus more accurately.
Sports photographers who adopted AF efficiently were those who came from a manual focus background and were accustomed to techniques that were proven to get more infocus shots. I used to shoot sports mostly manual focus and started a little bit of AF with the f4s, but it wasn't up to the grade the F5 AF provides. You'd learn the sport and players well enough to predict when they'd be making a move and you'd be prefocused for that. For basketball, soccer, or football, follow a player instead of the action for a bit and they will follow relatively predictable patterns that you can focus easily. I.e. in basketball, a defender will be paired up against an opponent and after a turnover or two you can figure out who's ordered to be on who. For track, pre-focus where someone will be at their peak in a long jump or focus on a marker on the track they will be running past or for long distance relays, they use the same point of handoff.
I would get a 20-35mm f2.8D which replaces lots of primes. Also a 35-70mm f2.8