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.
I believe the OP has a 1D4 so:Maximum f/stop: Up to f/2.8
With the 39 AF points indicated by , high-precision, cross-type AF (both horizontal- and vertical-line sensitive) is possible during manual AF point selection.
During automatic AF point selection, the number of cross-type AF points will decrease from 39 points to 19 points. The remaining 26 AF points will be horizontal-line sensitive.Maximum f/stop: f/4
High-precision, cross-type AF with the center AF point is possible. The
remaining 44 AF points will be horizontal-line sensitive only.Maximum f/stop: f/5.6 or f/8
With f/5.6 lenses, all the AF points will be horizontal-line sensitive only.
With f/8 lenses, AF will be possible with the center AF point being horizontal-line sensitive only. AF will not work with the other AF points.
The question is - does the 70-300 show as an F/4 or not?