Well, we homeschooled our children, having first tried to mainstream them into the CA school system. We found that they had had educational experiences less structured than the one the NYT is here castigating and we also found that it took at least a year to get them back on track enough to actually have lessons and learn things.
So context is everything and the NYT is nothing, as usual.
Plenty of NYC public schools provide a less structured experience than this "unschooling" family, though with quite a bit more exposure to violence and drugs, and much less academic achievement. I refuse to worry about the shortcomings of particular homeschooling approaches until AFTER the public schools, which are spending about $20K/year/student of taxpayers money, are consistently delivering a decent education. It would be misdirected worry.