In my opinion, Cesium's only requirement for WebGL is a bit optimistic. In a perfect world of modern hardware it would run mostly without problems.
However, I've been testing it in my environment with a variety of older hardware pc's, laptops and even an older generation panasonic toughbook.
For example, on a Windows 7 PC with Radeon 6450 HD, this is how I see ground primitives :

On another machine with an Intel onboard graphics with a WDDM driver (which was blacklisted), I had troubles with Billboards and Points - their textures had weird buggy colors. But blacklisted GPU are not trustworthy.
On my Ubuntu machine I get half FPS than on Windows with a modern low-end nVidia card on a monitor with a resolution of 1920x1080. I guess it is related to the lack of proper drivers for ubuntu.
On an old toughbook I can't get decent framerate when just hundreds of objects.
So to me it is really not only having WebGL but the specific GPU's capabilities for some features, to be able to deliver an application with a good framerate and visual experience.
You can tweak Cesium to perform slightly better, like disabling fxaa, atmosphere (gave me twice fps boost) at the cost of quality.
All those are not Cesium's fault of course, but the inability of older and weak hardware to deal with it. Like games, you can't run them properly on older pc's even if the minimum requirements are met