I didn't read or follow the previous threads, so no idea if this has been mentioned or suggested before. But have any of you with the issue tried enabling anti-aliasing on transparency in the control panel? That setting is off by default. I don't think it will do anything for the shadows, but may help with the other aliasing issues.
As for the video for Heroes of the Storm I think it was? That's just how the game works, Diablo III has the same aliasing in the menus, they have it disabled there for whatever reason. The overwatch part of that clip is the transparency AA thing, which would help with things like fences and sometimes hair and foliage/trees etc.