As your question is really about AF:
Keep in mind that the 70-300 f/4-5.6 will not make use of the increased sensitivity f/2.8 center AF point. That means that the 70-300 f/4-5.6 will probably be a slight downgrade from a 70-200 f/2.8 mk2 AF-wise (and also from the prime, if its AF is as good as the 70-200 f/2.8's - I've not read much about it, it's way out of my reach). However, the 70-300 f/4-5.6 AF is still said to be quite good.
For the level of performance that you require from the AF, I would think that the only way to check if it is really up to YOUR needs, is to rent one and test it in your kind of situations. Users showing single easy-AF or even difficult-AF shots that ended up great on a forum like this, is no way to answer your question. You want the first image bang on with AF, preferably all the time, and I'm not sure if the lens will do that as well as the 70-200 f/2.8 II does.
As said, IQ-wise it's theoretically a very slight downgrade, but probably not noticable unless you know beforehand which is which. The thing you will notice is the DOF if you were shooting wide-open.