You're right, a phone call goes a long way in these situations. It's simple, but I'm guessing that training is limited so people just don't know the protocol.
They require that the parent that purchased the ticket for the child drop be the same person that drops the child off, presents ID, and fills out a ton of paperwork at the time.
Similarly, you must designate a person to receive the child at the other end, and that designated person is the only one allowed to receive the child (even if they have an emergency or something), and they must also present ID.
I had a hard time because I used a nickname for my designee (whom I've only ever known as 'Lynn Cooper', when her legal name was 'Jerry Lynn Cooper', and her driver's license showed 'Jerry Lynn Cooper'. They refused to release her initially and it took three calls to untangle and I, the father, but not the person who dropped her off at the airport initially, was not allowed to verify that 'Jerry Lynn Cooper' was the same 'Lynn Cooper' designated, despite being on a conference call with all parties.
So yeah, if we have to go through all that, they should probably be held responsible for making a phone call on a reroute.