Which may seem to make a lot of sense, since taiji practice is slow. But when you think a bit further, and remember that taiji is not an exercise but actually a martial art, this linking of breathing with action seems to lack a sense of practicality.
My teacher, Mr Kwek, has always told me that in practice, just breathe naturally. Think a bit more and we can see why. In a fight, you need to be able to open and close, draw in or strike out, regardless of whether you are inhaling or exhaling. Things are going to be happening a lot faster than during practice, and there is no way to match your breathing with the speed of actions. Your opponents are not going to wait for you to inhale before they come at you, so you need to be able to punch and strike even when inhaling. Things cannot be inhale (wait), exhale (punch), inhale (wait), exhale (punch)... You need to be able to take the most appropriate action, at the most appropriate time, whether you are inhaling or exhaling.
Another thing is that different people have different lung capacities. By controlling your breathing when practising, you may end up either breathing too fast, or holding your breath in between actions. And holding your breath is especially bad, since it tenses up your body, which is one of the things that you should never do in taiji.
Oh, and in pushing hands, if you link your actions to your breathing, you are giving your opponent an extra source of information to know what you are trying to do.