I would be shooting Football, Baseball & Swimming some motorsport. Just one body. Shooting wildlife would be as important,
Well, it's going to be tough to do everything with just one lens. Or I should say it will be tough to do everything WELL with one lens. But since you're considering the Canon 300mm 2.8 that leads me to believe you have $4000 to spend so we could be in business.
For wildlife and motorsport, reach is going to be critical. 100-400 would be the lens I'd suggest for those two purposes.
Swimming: 70-200 2.8 lens is the way to go. You may also require an external flash depending on the light levels in your given natatorium (and whether they allow flash). But more important, you need to be shooting from the pool deck not the stands if you want quality shots. If you don't have access to the pool deck you'll have very poor angles and even if you have enough reach, quality will suffer.
Football: you didn't indicate the level of play - midget, high school, college. Will games be day or night or both? Do you have access to the sidelines? As with swimming it's critical you have sideline access to get decent shots. If you don't have access, it isn't worth spending large $$$ on glass - your angles will be poor and too much of your view will be obstructed or you'll simply be too far away. If you're talking little kids you COULD get away with the 70-200. At high school level 300mm is a huge bonus and 400mm is ideal. Think of it this way - from your shooting position, a 200mm lens is good for about 25 yards of quality coverage, 300mm lens is good for 40 yards and a 400mm lens is good for 50-60 yards. 2.8 lens is ideal in all cases for subject isolation, but if you have night games it's absolutely required to get necessary shutter speeds. Also, if you're shooting night football you may find it necessary to use an external flash - even with 2.8 lenses. If you do I would suggest using a flash bracket and preferably a flash with external battery pack. Without the pack the recycle time on the flash will limit you to 1, maybe 2 shots in a burst.
Basball: Again you need to be shooting from the field to get quality results. Once again, focal length depends on level of play. t-ball you can use a 200mm lens. By the time you get to high school you need a 300mm lens to cover the entire infield. Again, 2.8 apertures are beneficial for subject isolation but not absolutely required unless you have night games.
Zoom vs. prime: No question prime lenses provide higher quality. But you're only shooting with a single body and you don't want to be swapping lenses in the field. So I would advise against prime lenses like the 300mm 2.8 for baseball & football (it's too long for swimming).
Third party vs. Canon: I have both Canon and Sigma lenses. Third party lenses come in all varieties - high quality down to utter crap. So, each lens has to be weighed and measured on it's own merits. To be honest I haven't seen much quality sports work (on canon mount) from other third party lenses besides sigma. There are 3 sigma lenses I would recommend for sports work: 120-300mm 2.8, 100-300 f4, 70-200 2.8.
I happen to own and use the Sigma 70-200 2.8 and the 120-300 2.8. Both are fantastic lenses. In the case of the 70-200, image quality is just as good as canon. Canon is slightly faster to focus in low light and has a little less purple fringing. In the case of the 120-300 vs canon 300mm 2.8 - no doubt the canon is quicker to focus and slightly sharper. This is probably canon's best performing telephoto. BUT, the 120-300 is about 90% of the canon. Sill outstanding and better than any other lens out there (beside the canon) at this focal length. AND it's a zoom which is critical. Shooting with a single body this is important.
So, what do I recommend?
70-200 2.8 for swimming (canon if you can afford it, if not sigma is absolutely a great alternative) ($1100 for canon, $850 for sigma)
Sigma 120-300 2.8 for baseball and football and motorsport ($2700)
Monopod for the 120-300 ($50)
Sigma 1.4x TC for extra reach for all outdoor stuff ($170)
For wildlife, the 120-300 can be a great lens - it takes a 1.4x well and I'm told a 2x very well. Another poster here, NHL, uses the lens alot for wildlife. But it's a rather heavy lens. Personally I still prefer to use my 100-400L for the wildlife work I do (which I admit is minimal) especially if I don't want to use a monopod. It really is tough to have one lens for both sports and wildlife though. For wildlife you need plenty of reach - preferably 400mm or more. For sports you really want wide apertures. So, I don't like to use my 100-400 for sports because of the 5.6 aperture.
Here are some sample shots with the lenses I've recommended.
Sigma 70-200 2.8 for swimming:
Sigma 120-300 2.8: