The requirment for producing I.D is not necessarily one of security. Its also there to stop me selling my Easyjet / Ryanair ticket (which for whatever reason I can not use) to somebody who does need it. Thus reducing their profits.
Rules are there in order to give staff members clear and concise instruction on the procedures for their jobs and to make for an efficient and safe service for all. However they are not totally inflexible and certain Managers must have the ability to be able to change / modify or ignore a particular instruction if certain circumstances dictate. If you cant trust your managers to do this then they shouldn't be in that position.
Whats wrong with the check-in person calling their supervisor to deal with the lady? That person can assess the situation, talk to a Passenger services manager or an Ops Manager and get a decision from there. Maybe the airline would have been happy to accept her under circumstances.