As to the back button focus questions . . .
Once BBF is setup the focus is not connected to the shutter release at all.
If you press the focus button the focus will engage according to the rules for the focus mode selected. If you release the focus button the focus will not change, not in any of the focus modes.
So, in a situation like yours you would BBF and release, recompose, press the shutter release. This is essentially the same as single shot mode but you don't have to hold the shutter release half way down while recomposing.
If you hold the focus button then the camera will continue to refocus. This is where some confusion comes in. What I've read, but the author stated that canon would not confirm or deny, is that the camera will continue to track focus even while the shutter button is pressed. With the focus and shutter on the same button focus tracking stops when the shutter is activated. This is a small differance but it has some important implications when working with narrow depth of field or fast moving objects. We are all naturally unstable and we sway. Holding the focus button in AI Servo allows the camera to continuously adjust focus to compensate for our movements, even while taking pictures. Again, based on your results, BBF and AI Servo would not have solved the focus issue you had. It's hard (impossilbe) to say exactly what happened and why the focus was on the bushes.
In your situation, where you wanted to recompose, holding the focus button would not work. The focus would be reset as you recomposed. In order to use BBF continuously, in AI Servo mode, you'd have to select a different focus point and put that point where you wanted to focus. In this situation you were shooting in full sun so you'd be fine using any of the focus points. The additional capability of the center point really comes into play in low light/low contrast situations.
Also know, it is 100% ok to shoot a bullseye shot and then crop it to give the desired composition. I used to think all those other photogs were so lucky or so skilled they always ended up in exactly the right place and time to get that perfect photo. Now I know better. Not that there isn't a lot of skill, knowledge, time, and hard work involved but there are also a lot of setup and post processing "tricks" to help capture those perfect shots.