That "first errand" is pretty much how I handled parenting, without the involvement of school/police/whatever supervision. I would lurk in the background while my kids went on their quasi-independent journeys of discovery, gradually increasing the radius of their area of autonomy proportional to their age and ability.
The other determining factor was indeed specific location. In our quiet suburb north of Houston, I didn't have a problem with my daughter playing unattended near the house for short periods. In a neighborhood in north Dallas with a MUCH higher crime rate, greater population density, and nearby high-traffic roads... we didn't even let her go to the mailbox in front of the house without a parent within arm's reach, and by then she was 8.
A later move to Cheyenne, Wyoming gave them nearly unlimited freedom. Our son would walk a mile or two from our house to the nearest school playground by the time he was 10, and 13-year-old daughter had no problem walking a couple of miles through the middle of town to get from a friend's house to the library.
Last time we visited Dallas, our vehicle broke down and I had to walk to a convenience store to get transmission fluid. My son - by then 15 - came with me, and with his physical training from karate and football I was glad to have him along to protect me.