Honestly.. (please don't take umbrage at this), the 'science' community has so many holes in it, it is reminiscent of a sieve. You should have a listen to all the interviews with the top Gluten scientists/researchers and professors from the Gluten Summit last year. Where does the quote from above come from? From the Science of testing link you put below.
You cannot take as given what anyone in the formal healthcare chain says. A good example is my son reacting to egg 7 hours after eating it when he was 7 months old. The NHS consultant said it could not have been an allergic reaction to egg as it was too long in manifesting (second exposure btw), then when my daughter had her near anaphylaxis to something she ate the A&E doctor said explicitly that it was likely something she ate whilst she was away from home with her Dad, which was over 36 hours previously. I questioned her on this and she said that it is very common in children to have this delayed kind of reaction... so who am I meant to believe, the consultant, who technically should know everything, or the A&E doctor who was younger and very aware of the constant discoveries in the field of IgE and IgG allergies. I actually said that I was IgG sensitive to a number of foods, to which she said... well that is allergy. And yet another doctor, on asking for my daughter to be properly tested for this compound, said IgG was also allergy. I have had two conformations in different forms of my IgG allergy to milk and since eliminating it have felt 100% better... one was pulse and the other was IgG protein sensitivity assay. My ex. boyfriend had a really rough day at Christmas a year ago, felt like he was going to have a heart attack. Two months later he did the IgG test and guess what he reacts to? TURKEY! Traditional medical science is very keen to lambast theories on IgG testing... and in a way I agree it is not 100% effective because if you eliminate a food you won't get a reaction - as my GP told me numerous times so it missed a raft of my sensitivities... which were picked up by the acupuncturist.
And as for limiting children's food.... I don't do that, I just go with my instinct with my two. If they avidly don't want to eat something, I think that often that is a sign they can't handle that particular food, if they have an allergic reaction that food is out, if they can't poop while they drink milk, I experiment by taking that out and challenging it. My kids are not great eaters, as I suspect and have yet to confirm that they both suffer from eating gluten, but I haven't cut it completely out of their diets yet because I don't want them getting the severe exposure symptoms that I get. Mother's gut instinct is very often right.