Stormy, all cacti do not have shallow roots. It completely depends on the species and there are 1000s upon 1000s of species of cacti. I am a manager at a specialized cactus nursery and I can tell you that some cactus species have massive root systems or tap roots. Cacti grow on all sorts of different terrain such as mountains (i.e., Cereus peruvianus), and without strong/deep roots they would simply fall over.
Moreover, it is absolutely correct that water is withheld until the specimen's soil has completely dried out. Cacti grown away from their indigenous region(s) always carry a high risk of fungal infections, root rot etc...A fantastic way to give them a fatal illness is by providing more and more water before their soil is bone dry. It is not against their evolution. Most cacti, with the exception of epiphyllums, evolved in areas with little to no liquid water. The good news is that hydroponic systems provide the correct amount of aerated water, so overwatering is not a concern.
Plants obtain nutrients from the sun and minerals in soil, which is what is in fertilizer/hydroponic solutions. I would use a system that a built in full spectrum LED light. You likely will want to raise up the temperature of the water and you can do so with a simple seedling heating mat. They are cheap and can be placed directly underneath the water tank. These mats typically raise temperatures between 15 to 20 degrees F above whatever the temperature is in the room. This heat will strengthen and promote faster root growth. You can grow cacti hydroponically without an issue. Thanks for listening! I know this was posted a while back, but I thought other people may come across it.