"But you're missing the point!"

I'm reading (and listening to) more fallout from our TV appearances, and I find myself saying "you're missing this point!" way too much. So, once again I try to go at it from the other side. What's the point? How do I say it more clearly so that others will hear it more clearly?

First, I hear a couple of times that unschooling (and homeschooling) parents are the ones who are more likely to be more involved in their kids' education, rather than less, just by the simple act of making a specific, and uncommon, choice on their education (and I even hear some of the critics agree with that point!). Next, I hear that "no kid will chose to learn algebra" or "how will they learn history" or "they need to get the fundamentals" before making their own choices. Then I hear that kids "don't know what they don't know, and need to be exposed to a lot of things" so that they know what they like and should focus on.

Too many people seem to make the following assumptions:

1. 'Kids make choices' means 'Parents do nothing.'
2. 'No arbitrary rules' means 'Free for all.'
3. 'No school teachers' means 'No learning.'

All of these assumptions must be accompanied by the assumption that the parents do not care about their kids.

So, since I'm a math/logic guy at heart, I ask that at least for a moment we accept the following assumptions:

1. Parents who unschool care about their children's education.
2. Parents who unschool want their kids to succeed.
3. Parents who unschool live in the real world, and are aware of all the concerns about unschooling and are aware of the rules and norms of society.
4. Parents who unschool spend a lot of effort and energy on their kids' education.

Let us then postulate that:

1. Unschooling parents want their kids to be exposed to as much as possible (assumptions #1 #3, and #4). Unschooled kids who are exposed to a lot must see some part of the world, and by extension, have to learn about it.
2. Unschooling parents demonstrate behaviors they want their kids to emulate (assumptions #2, #3, and #4).
3. Unschooling parents will seek out what they believe is the best for their kids, and will take advantage of all forms of learning, including classes, books, lectures, programs, other experts, and whatever else they need to get their kid whatever s/he needs (assumptions #1-4).
4. Unschooling kids who are using many forms of learning, and who are exposed to many things in the real word must live by the rules of society (and the museum, and the movie theater, and the bookstore, and the library, and so on...) (assumptions #1-4).

Is this a stretch? Is the acceptance of my assumptions an unreasonable request? If not, are the results of those assumptions too much to swallow? If not, then is it at least likely (maybe as much as the school system) that the kids will in fact learn the basics, see a lot, get the opportunity to find their passions (even though they may change over time), and that they will be ready to enter the real world as productive adults?