It’s not scattered in all directions. If you’re orbiting earth - and you can check this out by looking at the international space station webcam the blue sky we’re used to seeing is completely invisible - form outer space looking in, there is no blue sky.
No, the blue is not shifted with more intensity than the other wavelengths. And it’s not really that abtruse. As photons from the sun arrive and collide with atmospheric gases, the gas atoms get a little momentum, and the photons lose a little of theirs - like ping pong balls. The angle of the path of the photon is deflected, as it’s lost energy it’s wavelength gets longer.
Because of atmospheric gas collision the entire spectrum shifted down - so invisible blues are shifted into the visible spectrum.
Since scattered light is the only reason the sky is illuminated at all (the airless lunar sky, as you no doubt recall, is black), what you see is blue.
Where is this blue diverted to?..What actually happens, is since the path of sun light through the atmosphere is longer, more photon/molecule collisions happen, the photons lose more energy and are shifted down, into red. (photons do not change speed like ping pong balls - light always travels the same speed - but their wavelength changes - the colour changes)
The sun is not normally yellow. From outer space the sun always appears white. Why we see a yellow sun on earth is because the atmospheric collisions shift the image of the sun down a little and it appears yellow. The red sun at night is when the image has been shifted down a lot more and it is now red (and there aren’t anymore invisible blues left to shift into the visible spectrum).
There is no lunar sky - something that is transparent by the grace of not being there is not black. But possibly one of the most important things to note from the scattering that causes the sky to be blue, and the sun redder at sun set, is that that energy which is missing from the spectrum has gone somewhere else. It’s the heat that makes the green house effect work. …People often get confused with these principles.
The bluish tint from glass is the same scattering effect - even the blueness of the sea.