Education is essential if we want to give children a chance at social mobility. Traditional schools have their problems, including those you list, but at least children taught in them can get some qualifications, and those qualifications give the children a shot at doing something other than the same as their parents.

If the only education the parents themselves got was walking a pony up and down a railway line, it's unlikely that they are in a position to effectively homeschool a child.

Homeschooling only works when the parents are educated. Pre-industrial homeschooling only occurred in wealthy families, or perhaps learning a trade - the closest most people got to an education was what they were told in church. Modern western homeschooling is the result of state-directed schooling bringing the average person up to the level of education of an 18th century nobleman.

Itinerant children in Europe are not getting anything like the education of an American child who's well-to-do parents decided to homeschool them. The boys learn the family trades and the girls learn to look after the household. Parents often have poor literacy and numeracy and are not capable of or interested in providing an academic education.

Whether or not that's enough is up to you, but the opportunities of children raised without anything resembling a modern schooling in today's society are limited. The failure to provide adequate education to the children of itinerant communities is a huge driving factor of their continued poverty.