That's not really a bike path, now is it? It's a painted lane on a road.
I've never been in a place where I had to take a "busy main road" to get around locally. Cross it sometimes, sure, which can be done at a light. This sort of highlights the problem. Over-protectiveness, by examining the worst that can happen, and the worst scenario.
I've been riding in a big west coast city for the better part of two decades, and I have definitely not been able to rely on dedicated bike paths (or even multimodal "trails") to get around. I mostly ride in the road, and often in the car lane rather than a paint-separated bike lane when I judge that the latter is unsafe (can happen for various reasons but mostly conflicts with cars turning right, pulling out, etc.).
Despite that, I've avoided bad accidents and generally felt pretty safe, but only because I survived long enough to develop an acute intuition for when drivers are about to do something stupid/illegal/dangerous. Previous and contemporaneous experience driving a car on the same streets was also very important. I would not trust these streets with my daughter's safety on a bike, and I honestly won't trust her to read the idiots' minds until she has some experience doing so from inside a heavy steel cage.
Riding within neighborhoods like the one I live in now is a different question, though some people do drive quite fast through here, and the size of the typical modern car makes clearances and sight-lines rather tight. I do see a lot of older kids out on their own walking, often to and from school, which I think is great.
I've been riding in a big west coast city for the better part of two decades, and I have definitely not been able to rely on dedicated bike paths (or even multimodal "trails") to get around.
Fair enough, but that's not what I said. I said locally, and neighbourhood. You elude to this later in your comment, but it seems unjust to reply, and refue my comment, while changing scope.
These are kids. They can bike locally to friends houses, or a corner store, or a non-central local strip mall, without hitting main roads.
Which you in fact agree with, later in your comment!
It doesn't need to be all or nothing, people. Biking in local neighbourhoods can be safe, and it can still be dicey on the main roads.
I think the problem with your model is that it conflates these two concepts, which may sometimes be appropriate but definitely isn't universal. For example, if you walk about 3/4 of a block east from my house you will hit a relatively busy arterial that is honestly one of the more dangerous roads I've ever ridden on, with broken sight lines due to parked cars (which also constitute a dooring hazard), lots of cars pulling in and out from side streets and parking spots, delivery drivers blocking the bike lane, a busy pedestrian crossing (again with very poor visibility), and cracked and rutted pavement. Just from their small stature alone, a child riding on this road would be in significantly more danger than an adult would, to say nothing of the skill and experience it takes to navigate so many simultaneous conflicts safely.
On the other hand, if you walk west from my house, you can go the better part of a mile through a quiet, shady neighborhood before hitting a main road.
Now what is "local" and what is "neighborhood"? Is the grocery store about a 3 minute walk east of my house not "local" just because to get there you have to at least cross this busy, relatively dangerous arterial? I don't think that makes sense. However, it's definitely not part of my neighborhood in the go ride around the neighborhood sense I understood as a child growing up in the suburbs. So in my view I am not changing the scope of the conversation, but rather trying to draw your attention to a distinction that may not exist in the locales you have experience with, but is important in some of those you don't.
Well this is why I stipulated only crossing busy main roads, and going to local, quiet strip malls, and otherwise being on side roads.
This added context to my statements about local, and neighbourhood, and all kids get directions such as "but don't do that!!", it's part of life as a kid. And frankly, an adult, though it be self prescribed.
Naturally each situation is different, but busy rodes have traffic lights, otherwise you might tell the youngster "never cross that road".
And that still leaves loads of low risk areas, which ypu yourself admit is viewed as local.
I see a lot of either/or in some topics, and local non-car travel seems to be a big one. People want to rail on about fewer cars, it's not safe, more bike paths needed etc etc. And I find that any conversation revolving around it being safe ... anywhere to bike, elicits these sort of absolutist responses.
"It's just not safe!!!!"
But the reality is, it is safe... just with conditionals, which I mentioned, and with instructions to child, and of course taking into account the specific child too.
And this is how to avoid helicopter parenting. You don't use a cell phone tracker, to make sure they obeyed, you don't follow them around, you give restrictions and work with that.
If your point is that biking locally is safe anywhere (excepting absurd scenarios) for children as long as they can be trusted to accurately internalize and unerringly obey a complex set of directives concerning which roads they are allowed to ride on, where and how to cross those they are not, and warnings of hazards they aren't experientially equipped to intuitively understand, then sure, I agree with you.
However, in practice, I think you will find that there are very good reasons you see a lot of younger kids out walking by themselves in neighborhoods like mine, but very few on bikes (unattended). If it just came down to "helicopter parenting", you would not see the former either.