Sorry, I can't help myself
(bout 30 seconds in specifically)
Need to also think about the way that different light sources disperse their light. So for example, in computer graphics we have the concept of a point light source or a global light source.
The latter is actually an impossibility, i.e. how do you produce perfectly uniform light that extends forever without reducing in intensity. All light must have a point source, and as it so happens because the mechanics of the universe has a habit of producing spherical things, the point source are often spherical in nature, meaning that from any angle you perceive a bright disc that reduces in intensity the further from the course you go.
Put these sources far enough away from each and consider the three dimensional nature of what you're seeing (two objects next to each other in the sky may in fact be billions of light years from each other) and it starts to become obvious why the sky as we see it does not appear brighter than it is.
Not to mention the fact our eyes only perceive a small portion of the overall spectrum that is actually out there.
As has been said, go somewhere that takes you away from sea level and pollution sources and you'll be amazed at just how bright the sky is! You can even just find yourself a field somewhere out in the sticks up north somewhere and not even go that high and be amazed!