When your photos are out of focus are they all front focusing?
For giggles try to only use Canon lenses. Rule out "tweeked" Sigma lenses.
I'll have to say borrow or rent another 5dmk3 if they are available. My 5d3 has been an incredible responsive camera with little issues you are having. Even my little prosumer 80D has been an incredible tool for ai servo that can easily track extremely fast spinning amusement rides.
What would be interesting is if you can borrow/rent a 5dmk4 or 1dx2 and shoot the identical scenario's as your 5d3. This would be a good indication of gear combo (your lenses) with a different body. If your getting similar blurry images I'd look at your lenses or also analyze your expectations of how fast AF should work in such short distance of fast movement.
Subjects far from the camera even wideopen can have that safety margin of more dof and breathing room. Even my OLLLLLLD awesome 5d classic was able to manage to document my niece's soccer game with 70-200 f/2.8LIS. AI Servo actually worked but as you know buffer and AF speed is ancient. My point is that AI servo with some long distances allows more margin of error.
I recall being on a learning curve when tracking kids running straight at me. Close proximity with even f/2.8 aperture using AI Servo can be some form of a challenge to a tracking camera. I will either use AI Servo or single shot AF and I've been very pleased with my results.
Currently my "difficult" big test I have is following my kids on a fast spinning amusement park rides. I'll say my 80D gives me alot of confidence while my "flagship" Fuji X-t2 with my current lens combo definitely cannot match my prosumer 80D/lens combos.
Another suggestion is to "SPRAY AND PRAY". See if your camera finally has time to compensate and actually catch the photos. If your child is running/walking quickly at you just hammer a tonne of shots. See if your 5d3 actually catches sharp images.
When I first read this thread I was shocked to be honest. The 5d3 has been a refreshing "better" Canon body that is a universal body for some sports action, family documentation, portrait/events work.
Hope you resolve this issue soon....
My suggestion is to read everyone's contribution in helping you out. Easier to just nod your head assess your ways of formulating a solution.
If you truly think it's your 5d3........ either trade it in or sell it.