At 300mm, even f/5.6 is wafer-thin DoF, and for best sharpness/depth you're stopping down to f/8-ish - that's the same whether it's an f/2.8 or an f/5.6. And, the VR will give you back that stop easily enough.
Well, I'll have to disagree on that one. This argument is simply wrong. As a rule of thumb, you get the best sharpness out of a lens when you use 1-2 stops smaller aperture than its maximum. So, for an f/2.8 lens, you'll get the sharpest results at around f/3.5-f/4, whereas with a lens with max aperture of f/5.6, then you indeed need to step down to f/8.
I had a Sigma 70-300mm APO DG f/3.5-5.6 as my first tele lens. I know it doesn't come near the Nikon VR equivalent in terms of IQ, but if I stopped it down to f/8 and used it at around 275mm, I did get some decent results. I took the plunge and went for the 120-300mm EX f/2.8, and I must say, that the difference in IQ, versatility, AF speed, contrast, built quality, and the pure wow-factor of a solid-colour background, was beyond my belief.
Fast tele-lenses are expensive because they do what they are supposed to. I never tried the Sigma 70-200 2.8, but if it's anything like my 120-300, then I would say that you will never regret going for a 2.8 lens. I would never hesitate using mine at f/2.8. The results are just great. And if I stop down to f/3.2, they are even better.
As for the weight, you do get used to it. I remember the first time I got out and about with the 120-300, at the end of the day I couldn't lift the lens to eye-level, and I could feel it in my arms the next day. But now, I can carry it for a whole day and never notice it's there, and I am talking about a lens that's near the 3kg mark. And the weight is something that I learned to welcome, because it helps to stabilise the camera. It's the same thing with holding an SLR and a p&s camera. Have you noticed how wobbly it is to hold a small camera, compared to the weight of an SLR? Well, its the same with lenses. I get consistently good results hand-held at shutter speeds as low as 1/150 sec at the 300mm end, due to good technique aided with the extra weight.
Have to warn you though, as you might have noticed with your wide zoom, it is a slippery slope once you got your first fast lens.
I would say go for either the Sigma or the Tamron. I've too read some great reviews on both, and you wont be disappointed.