I'm not really sure that there are any mushrooms that just live on "soil". Mushrooms are not plants. While they do extract minerals from soil, and oxygen from the air, they need organic matter to survive. (I do usually think of "soil" as being rich in organic matter, and "dirt" as being devoid of it, but there are so many kinds of soils and dirts out there that it's impossible to lump them all together.) When I find mushrooms growing on the ground, if I dig down, they are often attached to a piece of wood buried beneath the dirt.
Mushrooms are usually divided into two camps, the mycorrhizal fungi, which are symbiotic with plants, and the saprophytic fungi, that break down organic matter. Among the saprophytic fungi, there are primary decomposers and secondary decomposers. The primary decomposers can digest plant matter (lignins and cellulose) directly, the secondary decomposers prefer their meal to be a bit pre-digested (
Coprinus and
Agaricus species, for example). King Stropharia, while being a primary decomposer, appears to need other micro-organisms to thrive, and thus appears to almost straddle the primary-secondary categories. (This is what makes it such a wonderful biofilter species, I think.)
If you are going to be using a straw bale as the basis of your biofilter, I think you
should use a species that grows well on straw bales. P. ostreatus and H. Ulmarius are probably the most aggressive in this regard, and thus the easiest to grow. Oyster mushrooms are easily obtained and easy to clone, and are thus well suited for beginners to grow. My own experiences with
Stropharia rugosu annulata suggest that it is a little bit trickier to grow on straw.
Psilocybe cyanescens is illegal to grow in many parts of the world. Stamets worked closely with government agencies to get permission for his experiments. Follow all
local laws in this regard.
In any case, a straw-bale biofilter is still pretty experimental at this point. Like any filter, it is going to have a limited lifespan - I'm guessing a few months, but there are a lot of variables. There is absolutely no reason not to try different species each time you replace the bale.
With a biofilter, you are not really trying to grow any mushrooms, just the parent mycelium, so it doesn't even matter if the species you are growing is edible or not. There is a lot of room for new discoveries here, and I don't think this area has even begun to be explored. This is still pioneer territory.