Maybe the question is not the best framing.
Maybe first ask your self just what we might mean by independent? It seems to me in the post you're subtly shifting towards freedom from external constraints, which I don't think is a fundamental aspect of independence.
Perhaps itemize your understanding of what criteria independence entails and then view that through the lens of degrees of freedom as the number or relationships (external constraints of a type) increases. I think developing the skills to navigate that problem space is one of the skills I see children needing to learn as part of becoming independent.
Tabooing the word "independent", the sort of situations I'm trying to talk about here are ones like, do you walk your kid to school or do you teach them how to do it and then stop walking them? Do you take your kid to the park or do you teach them how to go alone? Do you make their food for them or do you teach them to make their own food? There are a lot of reasons why you might choose one or another in various situations. Age is a huge factor (a 2yo shouldn't walk to the park alone even though they can now walk; a 12yo shouldn't need you to walk them to the park) but of course isn't the only one. In this post I'm writing some about how I think of these choices on the margin.
Thanks and I clearly missed the target of your posting. I sidetracked into the issue of how to chose one's one preferred alternative when external constraints might be present that amount to choosing a lower valued immediate return rather than an longer term value.
I am a fan of teaching to fish but also knowing when that can actually be done. The later is clearly very important.
When I talk about teaching my kids to be independent, trying to get them to where they can do more on their own, one response I often get is, paraphrasing:
I like spending time with my kids and don't see it as a burden. Bringing them to school, playing with them, making their food: these are all chances to connect and enjoy being together. They'll be old enough not to need me soon enough, and in the meantime I enjoy bringing them to the playground.
So, first, I also like spending time with my kids! We do a lot of things together, and I'm happy about that. But it's also common that they'll want to do things that I can't do with them:
One wants to go to the park and the other wants to stay home.
One of them is ready to go to school and wants to get there early to play with friends, but the other isn't ready yet.
With a third child now this comes up even more. At times:
The older two want to go over to a friend's house, but the baby is napping.
I'm still feeding the baby breakfast when it's time for school.
The best time for the baby's afternoon nap conflicts with school pickup.
The alternative to doing things on their own is typically not us doing the same things together. Instead, it's at least one kid needing to accept doing something they like much less, and typically a lot more indoor time.
I do think there is some truth in the original point, though. There are times when the alternative to "they go to the park" is just "I take them to the park". Sometimes that's fine (they want to play with friends, I want to write), other times less so (they want to play monster and I don't have anything that's actually more important). With this approach you do need to be thoughtful about making sure you're spending an amount of time with them that you all are, and will be, happy with.