Why does building with a bog take up such a huge area in comparison?
Effective bioconversion requires considerable surface area be available for the colonization of nitrifying bacteria. The amount of surface area required is dictated by the total volume of water being filtered. The filter media used in manufactured biofilters (bioballs, biomats, lava rock,etc.) have considerably more available surface area than the gravel that is used in construction of what many in the industry call 'Bogs'. Hence, to achieve the same amount of bioconversion a 'Bog' must be proportionately larger than a Biofilter.
The main reason, however, for constructing this form of filtration is to make space available for the use of plants (Phyto-filtration) and to create a habitat for myriad beneficial invertebrates.
On a side note, The term 'Bog' as applies to this type of filtration is incorrect. A Bog is a form of wetlands that is composed of spongy, decayed plant matter that is noted for it's high acidity. The filtration modules that are being constructed in the pond industry are more correctly identified as constructed Marshes.
I also don't like using the term "Bog' because it is British slang for the Toilet, Crapper, Latrine et.al.