we're given the e mail address of the check in desk so my daughter could send an e mail giving permission.
That's definitely flawed! Anyone can set up an email account in any name in a few minutes with no verification required!
Two weeks ago I flew from the UK to Spain with my 7 year old god-daughter (a very pretty little blonde girl) absolutely no questions asked at either end. Of course I had a notarised letter from her parents, in English and in Spanish, but nothing was asked for.