While not defending TL, their staff are clearly trained in spotting certain situations. This could have been an abduction, trafficked, underage sex through grooming or anything, so while embarassing to the customer and his daughter, the child's safety was the priority. TL are damned if the do, damned if they didnt.
As for providing proof of age when booking, this can easily be faked and needs to be checked on checking in.
Problem is what ID is suitable if child has no passport? I know of adults who cant open a bank account because they have no passport, no driving licence.