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In short, amateurs were able to outperform professionals in the economy because the amateurs did not take on tasks beyond the capability of any human being or any manageable group of human beings.
Amateurs outperforming professionals might mean that

(a) professionals do so poorly, it isn't hard to best them.
(b) amateurs do so well, making professionals obsolete.

Which one do you think is the case? Homeschooling children not just for minimum standards but for excellence and high achievement is a task that is nearly beyond the capability of a family with two wage earners. Homeschooling that feeds on an "I can do it myself" attitude may easily deprive itself of one of the lessons of economics that even Sowell must know very well: division of labor.

In short, amateurs were able to outperform professionals in the economy because the amateurs did not take on tasks beyond the capability of any human being or any manageable group of human beings.

Amateurs outperforming professionals might mean that
(a) professionals do so poorly, it isn't hard to best them.
(b) amateurs do so well, making professionals obsolete.
Which one do you think is the case? Homeschooling children not just for minimum standards but for excellence and high achievement is a task that is nearly beyond the capability of a family with two wage earners. Homeschooling that feeds on an "I can do it myself" attitude may easily deprive itself of one of the lessons of economics that even Sowell must know very well: division of labor.

First, I'd note that Sowell's discussion of the "professionals in the economy" referred in part to the socialist planners who presumed to dictate priorities to the people in the sense of setting tens of millions of prices. Do you think yourself, or anyone you have ever heard of, to be smart enough to know the relative difficulty of making 20 million different goods? "The market" sets those prices by an ongoing bidding process - the man who is most confident in his ability to make copper efficiently is the one who sets the limit on the price of copper, in conjunction with the man who wants the most copper. The man who is most confident in his ability to make steel and the man who wants the most steel, likewise - and so on and so forth throughout the economy. The "professional" assays to set the prices as fixed quantities - if only because he doesn't have the capability to do anything else - but the market prices may, depending on the commodity or good, fluctuate quite uncomfortably based on news.
Sowell also did talk about homeschooling. I think we can agree that nobody can teach much of anything to any person who doesn't have a motive to learn. And the motivation to learn tends to correlate very powerfully with parental involvement in education. The notorious "inner city" schools have notoriously bad results because so many of the students' parents didn't like school themselves, and don't establish expectations that their children will like it any better or will excel in learning anything. OTOH the very definition of homeschooling parent is an adult who expects his (or mostly her) children to do very well (at least in the context of their handicaps, if applicable) in learning.
If professional teachers can't teach without parental involvement, and if homeschooling parents are the epitome of parental involvement, the pedagogical virtues of suppressing homeschooling are, it seems to me, close to entirely illusory. A parent who is not actually competent to teach a subject almost certainly will know it. And if they know it, they will arrange for some other teaching of that subject - or abandon the homeschooling project itself. If they abandon the project, they presumably will have learned more respect for the professional.