The issue is that TSA is wrong. There are kids getting caught in the N-F-L net every day, and the only real way that you can proactively avoid it is to register the child's birthdate with the airline.
My kids are frequent fliers. They have mileage accounts, and for that reason their birthdates are on file with several airlines and have been for their entire lives. In that 12 years we have not had any problems. I'm an identity theft victim (HR at work got hacked), so I do check and safeguard. Having that information on file with the airlines has made travel a lot easier for us, and I personally find the convenience and the travel rewards to be worth the small risk. (Also, we travel internationally, so their passports are seen by a lot of people as well.)
The thing is, you really cannot perfectly safeguard that information anyway. It is out there and pretty easy to find if you are determined and have the proper tools. The same people who would be likely to steal it from the airlines or from TSA are just as likely to get it from a school system, a health insurance company, or the Social Security Administration. I can guard against dumpster-divers pretty effectively, but I'm kidding myself if I think that I can foil a determined hacker. My solution is just to be watchful, so that I can take effective steps quickly if a problem arises.
In the reading I've done, and my conversations with law enforcement since becoming an identity theft victim, one piece of information keeps getting repeated. As with physical abuse, kids are more at risk of identity theft from people that they know. Kids' identities are most commonly stolen by members of their extended families. Professional thieves would rather go after adults who have established favorable credit ratings already.