First of all, you may want to rethink the "professional (from the stands)" requirement. Many professional sports venues simply don't allow long lenses. Here, for example, is the policy from the LA Galaxy:
The following is a partial list of the prohibited camera equipment:
Flash photography Monopods or tripods Cameras with interchangeable lenses Cameras with lenses greater than 3½ inches in length
I once took a 70-200 into Gillette Stadium for a Patriots game. The security guard at the gate said I could take it in but cautioned me that if they saw me using it, it would likely be confiscated. So it stayed in the bag the whole time. I'm sure the same would have applied had it been a Revs game.
So I would check before assuming I could bring even a 70-300 to a stadium.
Now, as regards shooting youth soccer, yes, a 70-300 would be my suggestion in the Nikon realm. The 80-200 and 70-200 lenses are sweet, but they are pretty short for soccer, even on a DX body and even on a youth pitch. (Unless the children are very young and on a very small pitch.) I shoot a fair amount of high-school soccer with an FX camera and a 400 f/2.8 lens, which is just a little shorter than a DX body at 300 mm. I sometimes wish I had a longer lens!
Unfortunately, in the Nikon line, there isn't anything better than the 70-300 that isn't well outside the "affordable" range for most people. (We are all breathlessly awaiting an update to the 80-400.)
I haven't used the >300mm Sigmas, so I can't comment except to say that for soccer, it would definitely be beneficial to have more than 300mm.