Well there's a few things to consider. First, flourish ferts works in the same way as miracle gro, jobes, osmocote, etc. So it's not 'organic', but it doesn't necessarily mean it's toxic in any way. If you eat only organic labeled foods then I could see why it would be given extra consideration.
Second, if you are going down that road, then you'd only put gravel safe for human consumption in the tank, along with fish/inverts that have only been fed with foods safe for human consumption their entire lives, and so on and so forth. You'd also have to grow every potential plant from seed to be certain that it hasn't had these inorganic fertilizers in it's system, since store bought plants tend to have some kind of slow release fertilizer pellet or ball around their root base.
Also, the aquaponics setups rely on a hard substrate material to hold the plants in place, since water is flowing over it they don't typically use any kind of dirt or soil.
