I think the reason why people worry about cell phones and not other radio equipment is that they are transmitting very close to the body. Remember, EM intensity falls off as 1/r^2 so proximity makes a big difference.
What that means is, of course, increased exposure. The kids are not going to give a way their phones, they'll continue using them. When there is no wifi, they will be using the LTE or 3G or 2G networks. When the base station is further away,the phone will transmit at higher power. The phone itself is by far the dominant source of RF power to people's bodies.
Thus, ignorant parents want to decrease radiation, and the very proposals they make would increase it. Whether that really has any impact, we don't know, there's no proof, or a good theory about possible mechanisms, just speculation, mostly based on fear and suspicion.