The way things work on consoles is approximately similar to Windows/Linux/Mac, except for these important distinctions:
1. the hardware is a known target, as such the shader compilers and other components are carefully optimized only for this hardware, they do not produce intermediate bytecode formats or make basic assumptions of all hardware.
2. the APIs allow injecting raw command buffers, which means that you do not have to use the API to deliver geometry in any way shape or form, the overhead goes away but the burden of producing a good command buffer falls on the application when they use these direct-to-hardware API calls.
3. the APIs have much lower overhead as they are not a middle-man on the way to the hardware, but an API implemented (if not designed) specifically for the hardware. For example Microsoft had the legendary Michael Abrash working on their console drivers.
4. the hardware memory layout and access bandwidth is known to the developers, and certain optimization techniques become possible, for example rendering to a framebuffer in system memory for software processing (on Xbox 360 this is done for certain effects, on PS3 it is heavily utilized for deferred shading, motion blur and other techniques that run faster on the Cell SPE units), in some cases this has other special implications, like storage of sound effects in video memory on PS3 because the Cell SPE units have a separate memory path to video memory and thus can tap into this otherwise "unused" bandwidth for their purposes of sound mixing.
5. 3D stereo rendering is basic functionality on consoles.
The article is making the argument that we should be able to produce command buffers directly and insert them into the rendering stream (akin to OpenGL display-lists but new ones produced every frame instead of statically stored).
It is also making the argument that we should have explicit control over where our buffers are stored in memory (for instance rendering to system memory for software analysis techniques, like id Software Megatexture technology, which analyzes each frame which parts of the virtual texture need to be loaded).
There are more subtle aspects, such as knowing the exact hardware capabilities and designing for them, which are less of a "No API!" argument and more of a case of "Please optimize specifically for our cards!", which is a tough sell in the game industry.
AMD has already published much of the information that studios will need to make use of such functionality, for example the Radeon HD 6000 series shader microcode reference manual is public already.
Intel also has a track record of hardware specifications being public.
However NVIDIA is likely to require a non-disclosure agreement with each studio to unlock this kind of functionality, which prevents open discussion of techniques specific to their hardware.
Overall this may give AMD and Intel a substantial edge in the PC hardware market - because open discussion of graphics techniques is the backbone of the game industry.
On the fifth point it is worth noting that NVIDIA Geforce drivers offer stereo rendering in Direct3D but not OpenGL (despite it having a stereo rendering API from the beginning), they reserve this feature only for their Quadro series cards for purely marketing reasons, and this restriction prevents use of stereo rendering in many OpenGL-based indie games, another case of consoles besting PC in functionality for ridiculous reasons.