Well the OPs posted problem was focus jumping to a branch or other image element instead of staying on the bird. If the bird is stationary then when using BBAF there can be some advantage to releasing the AF-ON which fixes the focus in that particular situation. Basically if you deactivate the AF system by releasing the AF-ON button then the focus can't jump to some other element in the frame.Unless I have misread this thread: I use BBAF, usually with single point AF and always continue to press the BBAF button rather than releasing it, as unless it is a large, still bird, there is often movement. Is the suggestion that Iit is better to release when it is a fairly still bird and if so, what is the advantage?
But like you, I generally keep pressing the AF-ON button to keep the AF system activated with a single point right on the bird's eye since small birds tend to twitch a lot or glance in different directions and if image size is large in the frame (magnification relatively high) those small changes can make or break focus on the eye. And of course all of this is in the context of stationary, usually perched or ground feeding birds, as I'll generally use Group or one of the smaller Dynamic Area modes for BIF but will obviously keep the AF system activated while shooting moving subjects.