Just out of interest, was the advice different in the 70's and 80's, when not so many people had cars, and couldn't so easily take children to hospital themselves?
Lots of people had cars in the 70s and 80s - it wasn't the dark ages!!
But, we also had more hospitals, i.e. in every small town there'd be a "casualty" dept, maybe not open 24/7, and maybe not fully equipped to modern A&E standards, but absolutely capable of xrays, basic broken bone potting, dealing with minor injuries from relatively minor accidents etc. It was only the most serious cases, i.e. bad car accidents, heart attacks, etc that would go to the "big" infirmary in the next town with a "proper" emergency dept that was open 24/7. Ambulance drivers made the decision which hospital to go to based on their knowledge of which services were available at each.
You'd also have GP surgeries were you could just go on with minor injuries and, shock horror! they'd actually treat you there and then without waiting 4 weeks for an appointment.
The reason ambulances and A&E depts are now so busy is that GP surgeries turned themselves into an office hours based "desktop" job and won't deal with minor accidents/emergencies , and that loads of smaller hospital casualty depts were closed down. Quite simply, anyone with "minor" accidents/emergencies has nowhere to go anymore except for the big centralised A&E departments.
I remember breaking my arm at school around 1980 - teacher sent me to school office with a friend. School office secretary put me in her car and took me to the local hospital a mile away, checked me in and left me there. By the time my Mum had arrived from work, I'd had an x-ray and was in the "potting" room. That was in a small town hospital that had a 9-5 m-f "casualty" department. It was the same "casualty" dept I went for an ingrown toe nail removal - sent directly by my GP!
We do seem to be going backwards in the name of progress sometimes with the NHS.