It might not be a true experience for you, but I think that very much depends on where in the country you lived and has a lot to do with family background, upbringing and circumstances. You were obviously brought up in a city, for me we were semi-rural
Quite a lot of it rings true for me, especially the amount of freedom to roam, climbing trees, walking, running and roaming with friends up to 15 mile radius or more, especially at weekends when not at school, and travelling on (abundant and cheap!) public transport solo or with friends, (although maybe not quite as young as 5).
All the schools I attended were local and walking distance. Senior school was over 2 miles away. There were school buses, but we always usually preferred to walk the two miles there and back regardless of sun, rain, wind or snow and ice. After being dropped off at infant school on the very first day, I was never accompanied by anyone thereafter. From the age of 5 onwards I walked solo to and from infant and junior schools, which were just over half a mile away.
I'm talking late 50s and into the 60s