It's difficult to understand the hostility some people have toward homeschooling. Even if someone doesn't care for it, it is bizarre to
insist on others not doing it (in some cases, governments insist so much that it is illegal to homeschool). Of course,
parents are the primary educators of the their own children. They may delegate that responsibility to others for certain subjects, domains, or scope, but the authority rests with them. The decision of how to educate is also a prudential one. For the rest of this post, I will use "education" in the narrower sense of what would fall within the scope of the school.
That being said, you cannot categorically judge either homeschooling or "institutionalized" education, as the quality entirely depends on the concrete situation. Both can be done poorly or done well. There may be aspects here and there that set them apart, where one is better than the other, but on the whole, in principle, both can be done well or poorly. Both can fail or harm the child.
Of course, to be able to evaluate the quality of education requires that we first have at least a sense, if not a definition, of what education is and what it is for. Immediately, this is where the trouble starts.
If you ask most people today what education is about, the most common answer I would expect is "to prepare you for a job". Primary education is to prepare you for university, and university is for preparing you for a job. Interestingly, this is not the traditional mission of education, which is perhaps best embodied by the classical liberal arts taught in the trivium and quadrivium. Their aim was to free the human person as a human person, and a human person is a rational animal. The classical notion of freedom is the ability to be what you are — human, i.e., a rational animal — which is quite different from the modern notion of being able to do whatever you want. This classical notion of freedom is the reason for the liberal in liberal arts. Now, the modern concept of rationality also differs from the classical, so even here we have divergence.
The point is that the liberal arts were distinguished from the servile arts. It is the teaching of the servile arts that would prepare you for the job. While the gains of a liberal arts education translate into benefits in all things, they were not per se for the sake of specialized work. Their value was not instrumental, even if they do have downstream benefits for the instrumental. This is like the difference between theory and practice. One seeks understanding, the other seeks to achieve some kind of subordinate or secondary good.
Now, as to why homeschooling is becoming more attractive, we need to consider the reality of education as it actually is today. I don't want to turn this into an essay, but a few big motivations are:
* the poor quality of education
* the alienating and hostile nature of many schools
* the hostile ideological presuppositions of an education system, often insinuated rather than explicitly commanded
As to how effective homeschooling is at correcting for these faults, that will depend on the particular situation, more or less. From what I understand, homeschooling parents will often meet with other homeschooling parents and draw from curricula that already exist for this purpose. Sometimes these parents decide to found school themselves (as we are seeing in some cases with the rising interest in classical education).