From what I have read about root crops and hydroponics, the texture of the
growing medium is important for good formation (shape) of the root (editable food). Basically for the root to grow with the same shape that we are accustom to seeing in the market, the growing medium should have as close as possible properties and texture as the soil they are used to growing in. That's not to say they will not be editable in any other growing medium, just much more likely to be deformed the less like actual soil it is.
As for using Roockwool, aside the difference in texture, I can't see it being cost effective. I can usually get 2 pounds of carrots at the market for about 2 bucks. The amount of rockwool needed to grow that quantity of carrots will surly cost much more than that, and wont be able to be reused. Even if you wanted to try to sanitize the rockwool for reuse, you'll need to tear it apart to get the root veggies out of it.
The idea behind the sand system was to be able to provide the right texture for good formation of the root (editable part), as well as be able to continuously harvest and replant the same space. Beneficial microbes will be necessary in the system though. Because as you harvest some roots will break off and remain in the sand (or any growing medium), then they will feed pathogens. So a combination of beneficial microbes, acids, and fungi will be needed to break down the dead roots, and keep pathogens from getting a foothold in the system.
P.S.
Green onions aren't really a root crop, but we like using a lot of them. I have seen where green onions have been grown efficiently in a
aeroponic type system as well. In the future I would like to work out a system for continued green onion production using aeroponic methods also.