Practical organic production seems to revolve around Compost. What's wrong with leaving the weeds, grass, seaweed or whatever to lie on the ground or under trees to break down naturally. It seems strange to remove the plant material from the area, compost it, then take it back, almost as if composting is supposed to mysteriously add some missing ingredient to the material being composted.
Practical organic production does revolve around compost. Health of the soil depends on the compost it receives and success in the garden depends on the soil. Natural decomposition occurs on forest floors and creating a compost pile intentionally replicates this natural process.
There is nothing wrong with leaving weeds, etc to breakdown naturally, this is actually mulching. Mulch is a mat of dead plant material which will conserve moisture, protect plants/trees against heat, wind and rain and suppress weed growth - however, mulch is not a fertiliser.
The correct ingredients for a compost pile are oxygen, water, heat, carbon (leaves, straw, woody materials); nitrogen (grass, food scraps, manures); micro-organisms and macro-organisms. These are all necessary to making a good compost. When matured it should have transformed into a crumbly soil mixture with a sweet, clean aroma.
Heat is the missing ingredient and there is no mystery. The larger the pile, the more likely the temperature will increase. Below 40 degrees celcius is the mesophillic stage whereby fungi and acid producing bacteria breakdown sugars and starches, and above 40 degrees, the thermophillic stage activates with the breakdown of proteins, fats, cellulose and hemi-cellulose. All this activity creates heat which at 55-60 degrees celcius will kill pathogens, parasites and weed seeds.