My LDS requires a nitrox card. I imagine most do. There are more ways to get dead with nitrox than there are with air within recreational limits, and very few shops want to be the one that killed someone. Even cynical people find the publicity unpleasant.
As Damselfish said, oxygen becomes toxic when under pressure. Also, oxygen may have a dangerous cumulative effect. Think of it like nitrogen loading - the danger may be function of the percentage of oxygen in your breathing gas, the pressure, and the amount of time at that pressure. Thus, you may have more than one time limit to track when diving, which complicates dive planning.
Also, the higher the oxygen content in a gas, the more likely that flammable materials will combust at a given temperature. This includes things like the o-rings and lubricant used in your breathing equipment. If you don't know your equipment and use gas that is too oxygen-rich for it, you could run an unacceptable risk of combustion, including potentially explosive combustion. Tank go boom.
Nitrox is mixed in the tank in a few different ways. One common way starts with pure oxygen in the tank, and then adds a gas with a known, and lower, oxygen concentration to the mix. Again, if your equipment is not O2 safe, things could get messy.
Even if it does not cause a fire, oxygen-enriched gas may react, particularly with hydrocarbons (think lubricant, again,) to produce carbon monoxide. If this happens in your tank, it is unlikely that you will detect it until you have symptoms of poisoning (which can include passing out, which is not recommended in 60 fsw).
It is technically possible to cause lung damage by breathing elevated concentrations of oxygen for a long period. You would have to work pretty hard to do it in SCUBA, I think. Also, it is possible to damage your retinas. I do not know how likely this is in SCUBA, but I think it is more likely in a hospital.
In short, nitrox can kill you in a number of ways that air is much less likely to do. Additional training is important. Because nitrogen loading is only one of many dangers in SCUBA, and because naive nitrox use introduces additional risks, it seems unlikely that the use of nitrox would decrease the net danger to your children. I will not let my son dive nitrox until he is certified to do so.
