XGI produced a series of GPU's, the Volari series
including the Volari V3, V5 and V8
and the Z7 series for Server applications
they are currently in the begining stages of generation 2's launch, the Volari 8300 and 8600, with the 8300 launched a few weeks ago...
the V8 and V5 single chip solutions provide admirable performance currently, inside of their actual competition ranges (the V5 targets the FX 5500, and compets with it, the V8 is targeted at the 5700 range, and can hold it's own)
XGI's issues are related to the dual GPU solution not being designed very well, and poor hardware AA support on the GPU's themselves (1 subsample per pipe, which supports the theory of the Volari V series being super pipelines)
Also, the S3 card does not manage to beat out anything overall, it manages to have floppy performance depending on the application. FSAA is terrible on the S2X series of GPU's, the GPU itself attempts to use an RGSS pattern which it doesn't achive through normal methods (it doesn't jitter the geometry, it rotates the image)
However, it is a very large step towards closing the gap to nVidia and ATI
Also, S3 wouldn't really classify as big, and it isn't a chip maker (neither is ATI, nor is nV, or XGI or any of those, they just design)
As to Intel and AMD competing, AMD has no chipsets produced for the consumer market, and only a handful for the server market, while Intel produces it's own chipets, and it's own integrated GPU's, the GPU market itself is small compared to the CPU market, while Intel can make more money by producing basic cores for integrated operation, than by desiging a more complex processor for powerful 3D...
Also, they'd be starting from nothing, and market image would be a serious question, would you really want to have a computer with an AMD graphics card? It just doesn't seem normal, and while AMD might design an amazing graphics card, there is always the question of marketability, in addition to AMD not really having a 3D Design unit (it's not the same as designing a CPU, well not entirely)