No, f.lux only tweens the color profiles of the OS in realtime to shift the white point. It's basically a wrapper around a function already in the OS together with a tween library. The only thing f.lux does is control a hidden API /9 it's not comparable to email or a browser.
Or are you suggesting f.lux passes the entire framebuffer at any given point through their code and adjust the color temperature using a shader on the GPU? (No.)
F.lux has nothing to do with GPU or shader. Its an application which tweeks display temperature controls. All modern OS provides such interface, on Linux, its x server extension, on Windows its GDI interface. So, its an application that uses the OS interface, that doesn't mean that it has to be made part of OS. At OS level, you provide functionality of underlying hardware but don't decide policies or features. f.lux is a feature not a functionality to display.
Or are you suggesting f.lux passes the entire framebuffer at any given point through their code and adjust the color temperature using a shader on the GPU? (No.)