5d3 works just fine for sports. I keep a wider lens on it and have no tracking/focus issues at all. 1Dx is nice (I get to use one now and again and plan on buying) but my 7D does just fine for 95% of my shots. The only reason I want a 1Dx is night skiing/snowboarding is a pain with the 7D as is any indoor. I keep a 8-15 f4 fish or 16-35 UWA on my 5dMK3 and on high speed have yet to miss a shot I want. Usually have a 70-200 2.8 on my 7d or longer depending on my location.
The most important thing.. rather then body or lens .. is knowing your sport inside and out. Where your subject will be at what time and the best location to get the shot. Knowing what base the play will be in baseball, seeing any advantage during a power play, sitting under the net on one end of a basketball court and using a UWA or Fish to capture that game changing dunk or that one point in the course that everyone has had trouble with but you know one skier is going to nail the turn and give you "that shot".
Note on the 7D... I have the 7D and I've heard some people say it's AF is better than even the 1DMK3, that is simply not true. I tried basically every AF combination on it and it just does not give as much keepers as 1DMK3. So if I were to go down the 1 and 5series to a 7, the MKII better be damn good in its AF area. Also the high ISO is just a pain to deal with. I know that knowing the sport I am shooting does benefit me;however, I do like AI servo and track players rather than focus trap at a given point.
I get a 80% keep rate with my 7D shooting skiing/snowboarding, Kayaking, Kiteboarding and some College Level Baseball/Basketball/Football. Not having used a 1DMK3 I can't compare but will say the noise on the 7D past a point makes it useless which is where I swap in a 5DMK3. I had the same issue before I sat down and learned how the tracking and servo mode worked on my 7D for different types of motion and subject matter. Now that I know I have custom profiles saved to my C1/2/3 on the dial for specific situations for easy changing on the fly and can go from keeping focus on a subject for half-pipe events to almost instant focus subject change for boarder/skier cross. My lens selection is a huge part what I do as well, since I am usually static. I use the focus preset a lot on my 200 f/2 and 400 f/2.8 MKII too when I know my subjects will be in two locations that I can get the images I want and will work for my client.