Despite my subject "telephoto Zoom" I think you have made a very good suggestion Taquin.
I noticed that I am using a focal length of 80-90mm with my kid's basketball and my problem has been the slow AF and f setting.
I will look at an appropriate prime.
Same goes for concerts (indoor orchestras ) where I usually set up my camera on a tripod 'down the back". I may find an appropriate fast prime with more experience.
You might want to take a look at the Sigma 70-200 f/2.8 EX HSM. About 600 UK pounds, so a bit more expensive than the Canon 70-300 IS, and a bit less expensive than the 70-300 DO.
It is a great lens... optically not quite on a par with the Canon L equivalent, but pretty damn close. It doesn't have IS of course, but in these days of sensors going up to ISO1600 and beyond, that may not matter so much. I have owned one of these for years and will not be parted from it: very sharp, great bokeh, very quiet, and very fast AF speed due to the big 2.8 aperture. Also the build quality is great - it can't be compared to the rattly and squeaky old Canon 70-300 f/4-5.6 which I sold recently. In most respects it is on a par with Canon L glass, for half the money. I believe that Tamron make a good equivalent too, but I haven't used one.
You may have read stories about compatibility issues between Canon and Sigma lenses. This is certainly a problem with older model Sigma lenses, but AFAIK there are no issues with Sigma's HSM lenses which they introduced a few years back.
Judging by your needs and your budget, this one might well hit the spot.