Hi,
MOSFETs are used in the output stage (at least) of the class D amplifier because they are very fast when used as a switching device and class D is basically a bipolar mode buck switching regulator that has an averaged response that can follow an audio signal.
As far as using them in a linear design, it's really up to the designer but again when used in the output stage it seems easier to get lower crossover distortion, if done right.
The design presented here just seems wayyyyy over complicated, and that tells me that maybe the design isnt that good to begin with anyway as i dont think anyone with a lot of experience in amplifier design would make such a complicated off the wall design unless it was for some very special purpose and a regular audio amplifier doesnt fit into that category.
The general rule here is that if you find a design that is much more complicated than most other designs you find for the same application, then there darn well better be something extraordinary about it to compensate for the extra complexity.
This extraordinary property can come as almost anything: a cheaper part, a more expensive but longer lasting part, a better high end response, a better low end response, etc., but if you cant find something great about it then it's not worth using the design.