One shot, so it isn't the focus re-adjusting it's self unexpectedly....
Well, there's your problem. With any moving subject you must use AI Servo.
One Shot is only for stationary subjects. It achieves focus, then stops ("locks"). If the subject (and/or the photographer) is moving, then by the time you trip the shutter it's no longer in focus. (Incidentally, if using a zoom lens, nearly all modern ones are "varifocal" designs, which means they don't maintain focus when the focal length is changed... So whenever you zoom the lens, it's necessary to re-focus if using One Shot.)
AI Servo is continuous focus, designed to track moving subjects. This is what you need to be using for BIF. (It also will correct for any loss of focus due to changing focal length of a varifocal zoom.)
Centre group focus (9 sensors) gives poor results, image doesn't focus accurately. nowhere in the zone is focussed, in fact nowhere in the image is accurately focussed. This is a problem. I was trying to photograph some birds - no, the flying variety and using the single 1pt centre spot it was obviously very difficult to accurately target a bird in flight. When I did hit one the focus results were good.
Using the centre group of 9 spots various spots would flash black squares as I tracked the bird, focus would lock and an image could be taken. The problem was the focus was very soft when zoomed in a bit.
Yes, it's more difficult to use a single point when shooting a moving subject, but it's the only way you can be certain where the camera & lens will focus.
If using a Zone, the focus is going to be on the closest thing that's covered by one of the active AF points. In the case of a bird, that's often the nearest wing tip... and depending upon depth of field (lens focal length, aperture being used and distance to subject), this can result in the bird's head and body not being sharply in focus. When I'm using Zone Focus or All Points/Auto Selection (which isn't all that often), I will usually stop the lens down a bit to be sure that there's plenty of depth of field, so that some minor focus error won't matter. If, for example, I'm photographing moderately large birds in flight, I might use f5.6 to f8, so that the entire bird will be in focus when the lens is focused on the nearest wingtip.
There is also a problem when using the A+ automatic point and shoot setting because it uses multi focus spots rather than the centre 1pt spot and so results are out of focus or "soft focus" to use the polite phrase.
That's correct... in the fully automatic A+ "point n shoot/snapshot" mode, the camera will only allow auto AF point selection, all points. It also forces the camera to use AI Focus, which is not really a focus mode at all. In AI Focus the camera is supposed to decide for you whether or not the subject is moving, then switch to whichever is appropriate: One Shot or AI Servo. I haven't tried AI Focus in many years, but last time I did it seemed to cause some slight delay and sometimes choose the incorrect mode.
A+ also restricts other things... such as only being able to shoot JPEGs.
Simplest solution, don't use A+ mode. Set the camera up yourself. Don't rely on the automation.
Or, if you must use it, learn to accept that the automation isn't perfect... It can't be with any camera since there are just too many variables in photography. The more complex the camera, the harder it is to fully automate it.
Coming from a camera with a simpler AF system, as you did, it will take some time to become accustomed to how the 70D's AF system works. I went through the same thing five years ago when I switched to 7Ds, which use a very similar AF system. I found that the "KISS" rule applies... Keeping it simple... i.e., using a single point most of the time and the "fancy AF pattern" modes only in very specific instances when they are most helpful... nets me the best results.
I suspect your camera is fine. It's just you learning how to use it, how it works. Spend some time studying tutorials on the 70D's AF system (or similar on 7D).
I went from 50D to 7D (your 600D uses almost identical AF to 50D, and your 70D uses almost identical to 7D)... At first it was a mess. Where I enjoyed 90% plus in-focus with 50D, I suddenly was only getting around 50% with the supposedly "better" 7D AF system! I almost returned the 7Ds and went back to the 50Ds. But I stuck with it and once I learned to use them, I'm now above 95% in-focus in all but the most difficult situations (low light/low contrast).
So stick with it. It gets easier and, eventually, better.
EDIT: Some settings you might try (as they are on my 7D.... your 70D will be a bit different, no doubt, so hopefully the description may help)....
Custom Function III, 1 Autofocus/Drive, AI Servo Tracking Sensitivity - This isn't quite what it sounds... It actually determines how quick the AF is to jump to a different point of focus, should there be a temporary object between you and the subject or even something behind the subject. Most users new to the system find it beneficial to set this slower, so that the camera is less prone to leave the initial subject. (Using Back Button Focus, with practice I can avoid accidental mis-focus by "dodging" the occasional object, by turning AF on and off with my thumb. Similar can be done, but isn't as easy IMO, when using the AF on the shutter release half-press. With practice, I turn this back up to normal or even slightly toward fast when I'm shooting moving subjects that change direction or move erratically.)
CFn III, 2 Autofocus/Drive, AI Servo 1st/2nd Image Priority - Has to do with focus precision when firing a burst of shots while using AI Servo. I leave it set to the default "AF Priority/Tracking Priority". This tells the camera it's okay to slow down frame rate or delay shutter release for a fraction of a second, that my priority is that the first shot be in focus, and that second and all subsequent shots also be in focus (vs speed of release, although focus accuracy might be compromised).
CFn. III, 3 Autofocus/Drive, AI Servo AF Tracking Method - Has to do with tracking in AI Servo, but only when using multiple AF points. If you want to allow the subject to move from one AF point to another, use 1. Continuous.... setting.
CFn III, 4 - I just leave set at the default "Focus Search On".
CFn III, 5 Autofocus/Drive, AF Microadjusment - I set to selection 2. "Adjust by lens". Your camera uses a newer, more powerful version of Microadjust... highly recommended to read up on it and use it.
CFn III, 6 on my camera simply enables or disables different AF Pattern modes (All Points, Zone, Expansion, Single Point and Spot Focus... Your camera doesn't have Expansion or Spot, I believe.)
CFn III, 7 - Merely determines how scrolling through the AF points works... whether or not it stops at the edge, or continues to the other side. Set whatever you prefer.
CFn III, 8 - Just enables or disables VF display illumination, or leaves it to the camera to automatically decide (default). Set as you see fit. This is the "red flash" telling you that AF is starting... and in One Shot, that focus has been achieved and locked.
CFn III, 9 Autofocus/Drive, Display All AF Points - I leave this disabled because I don't want all the black boxes to display all the time. I only want the selected pattern and active AF point to display.
CFn III, 10 Autofocus/Drive, Focus Display In AI Servo/MF - I leave enabled (default) because I want to see the AF point to be sure I'm targeting the subject well. Some might turn this off if they find the display distracting, such as in one of the highly automated modes. But I use those so seldom, and I want to see those AF points to put them on target!
CFn III, 11 - has to do with using the AF-Assist feature of some flashes. I have this set to only do so on external flashes, because I don't use the built-in flash for various reasons and it's AF-Assist is done with a series of bright white flashes... while external flash uses a much less intrusive projected near-IR grid.
CFn III, 12 - Autofocus/Drive, Orientation Linked AF Point - This is an important one. I leave it set to 0. "Same for both vertical and horizontal" (default). I tried it set to allow different selections for horiz. and vert., but it's a bit tricky and I noted a slight delay every time I rotated the camera, while it oriented itself. If you choose to use this, you need to be aware of a couple things. First, there are one horizontal setting and two vertical... vertical with the grip at the top and vertical with the grip at the bottom. If you set these up, you need to remember to set up all three, or the camera will act wonky. Also, be aware that when using this you not only can set different AF Points, you also can choose different AF Patterns, depending upon orientation. If it weren't for the slight delay while the camera is "thinking about it", this might be useful to have a quick way to change from one AF Pattern to another, merely by flipping the camera to another orientation. However I haven't found it practical. OTOH, I haven't used it a lot, so might get more adept at it, given more practice.
Cfn III, 13 - is just the old Mirror Lockup setting. I always hated that Canon buried this deep in the menu. But on the more recent models with Live View, it doesn't really matter since LV does essentially the same thing, and uses a direct access button right on the back of the camera.
Back Button Focusing is something you might want to try, if you haven't already. It's particularly popular among sports shooters, air show photogs... and birders, especially for BIF. It's super easy to set up on 7D and 70D. All you really need to do is go in and remove AF from the shutter release button. BBF is already assigned to the AF-On button on the back of the camera (I go a step farther and swap the function of the AF-On and */AE-Lock buttons, just because I prefer to use the larger * button for focusing... but this is optional). BBF, if you haven't used it, puts you more fully in control of when and where the camera focuses. It's particularly helpful with moving subjects and AI Servo, allowing you to start focus well in advance of taking the shot, so that it's sharply in focus when you trip the shutter.
BBF takes a little practice, but soon becomes second nature. Once learned, most users never look back and some even wonder how they ever got by without it!