AOG - I too remeber the 'good old days' when such situations could be resolved in the way yours were, and i am sure you join me in bemoaning their passing.
I think the lady was probably flustered and confused, and thought of the AA as a first response - correctly as it turned out - because their standard proceedure would have been to assist this lady and then sort out membership - which is standard for offering their services.
I agree that had the lady called out the police or fire brigade, she could have been accused of wasting public resources.
I recall years ago calling the fire brigade at 11:00 p.m.out to turn of our street stop cock because a kitchen pipe in the celing space has burst and was flooding the kitchen. The officer in charge of the crew advised me that this was considered a 'non-emergency', and they may charge me a considerable fee for their service. They didn't charge, and am unsure if that system still works - but who would want to test it in our modern financially straitened times?
With the hindsight offered to us sitting at our keyboards, we all have ideas of what the correct action is / was, but as a mum with a crying toddler, who knows what we would have done?