I've had the Sony 70-200 f/2.8 for several years - got it as a BD present from DH after I got my Sony A700 - and really like it a lot for sports. Personally I have not found it too heavy to use at any time. Now the 70-400 - that's a lens I have found weight to be an issue. If you're shooting some type of sport that is somewhat stationary - say gymnastics which has quite a few apparatus or perhaps tennis or something similar that is fairly contained then a prime could work but otherwise I'd find it pretty frustrating to not use a zoom unless you would definitely be far enough away that your fixed focal length would never be too great.
Also, about the Sony 70-300 G, I love that lens! It's been my favorite go to zoom since the moment I got it and I use it even more than my 70-200. If the light is good enough then that's what I'll take.