As to the weight, when many of your lenses are large aperture you get used to it and the mid range aperture lenses feel very light.
Like having a 24-105mm on the 5DMKIII feels a lot lighter than the 24-70MKI but you get used to it, so for a zoom of that magnitude 1050g isn't a lot. Compare that to the 70-200mm MKII which is 1490g add a 2x converter 325g and a mid body 860g = 2675g compared to mid range body and 70-300mm = 1935g. That weight saving is the same as taking a 70-200mm F4 IS also. So you could take another high end piece of glass to fill another gap like a wide angle.
I get the general point but I'm not sure "you get used to it" is really the issue. As with just about everything to do with camera gear, it's about each person working out what set of trade offs suits them best for what they want to do. Looked at in isolation, I agree 300g is not much at all ... but you can say the same about the 740g difference in the above example - it's really not that much. On the other hand, if you're carrying something far enough and trying to move fast enough, etc, you'll want to shave 300g off every item you can. In my case I'm sometimes (sadly not often - too much time doing a desk job) travelling with people who aren't so interested in photography so aren't going to cut me any slack, and we're covering quite a lot of rough terrain. I've done it with a 7D, a 70-200 2.8, a (Sigma) 24-70 2.8, a wide angle zoom and other bits and pieces and lived to tell the tale, but the weight made it all a bit less enjoyable. (And I'm someone who spends my weekends doing things like running up hills with a 20kg+ pack on my back and pushing heavy sleds in the name of fitness.)
A couple of years ago I looked hard at moving to m4/3, but in the end decided to go the other way to full frame. But I chose a 6D rather than a 5DIII, and apart from my 70-200 2.8 all my other lenses now are at the light end for lenses in their class. Saving a bit of weight on each of a number of items adds up in a way which matters to me - and means I enjoy using the gear I have. The OP may or may not have reasons (not necessarily the same as mine!) for valuing a weight reduction even if it's only 300g.
Anyway, my only real point is that there are a lot of factors which play into what's the "best choice" for someone. I completely understand why some people prefer the 70-300L - the extra reach being the most likely reason, but for some even just being able to carry the lens vertically in a bag might make it worth it. But there is nothing wrong with choosing the 70-200 4 IS either. And there is nothing wrong with deciding 300g extra is not something you want, even if it's something you could get used to.