Good last point there...which is why I said Bertoni said it better than I. And I should have said "bio media raises the nitrate level in your tank".
Liverock absolutely does harbor anerobic bacteria - which reduces nitrate, just very slowly. Bio media generally uses aerobic bacteria...which increases nitrates.
Well yes, actually, I guess just not as much as generally necessary, and not in comparable numbers to the others. Hence whilst there is some nitrate reduction, in general most still see nitrates increasing nonetheless with live rock.
But anyways, just be careful how you refer to biological media, because it encompasses a large diversity of products. For example, that holder that people get for macroalgae is actually a biological filter media, as it promotes biological filtration through growing macroalgae within itself. Not what people normally think of, but yes, it is in fact a type of biological filtration, and the media to allow its controlled growth is biological filter media.
But anyways, in terms of microbial biological filter media:
Matrix and de-nitrate can also harbor denitrifying bacteria, at a high enough level. Besides Marine Pure, that is. There's a few other products out there that is capable of harboring a relatively large amount of denitrifying bacteria. Sand, of course, is also a potentially major biological filter media that can harbor denitrifying bacteria.
Although the majority of biological filter media do indeed not be able to harbor a large amount of denitrifying bacteria, as, well, one could put a toothpick in the tank and call it a biological filter media, it's not good to blanket-term biological filter media as nitrate factories, because that's too much of a generalization. People will get confused. 'Wait, Marine Pure can lower nitrate, but it is a biological filter media, shouldn't it be a nitrate factory?' would be what happens.
But that's my opinion anyways. You have yours so if you don't agree, that's cool.
As for artificial biological filter media 'do seem to cause a higher nitrate level in some cases', well bertoni gave one reason - inhibition of denitrification, mostly through not creating anaerobic conditions. The other reason is that some artificial media simply is able to harbor much more bacteria than live rock, for example, however this simply means that ammonia and nitrite is being reduced much faster, which just means more nitrate is produced much faster. Either way it is about not having the amount of denitrifying bacteria needed to match the amount of nitrifying bacteria.
