That time line for factory workers is too general in nature and it leaps over time, perhaps by half a century or more. The fact of life in the early 19th century was if you were not rich, you had to work and it was going to be manual labor except in rare instances because of the nature of the prevailing economy. Laws required whites and free Blacks to be indentured to learn a trade after age 12 because there not free public schools available yet except in some churches, such as the Lutheran and Presbyterian churches. If they had money, this could be avoided by enrollment in a private school or they could be demonstrably employed in the family business to escape the indenture requirement (speaking specifically of New York State and England, but other states were similar, the requirement was similar to our present laws keeping children in school to a certain age or grade).
Unions arose after slavery, after the Civil War. Some factories in thst period were not "sweat shops", New England textile mills hired young women who likely could read and write, housed them in dormatories and regulated their conduct. Hours were long, pay was low, but we were emerging from what was almost a barter economy at the family farm level outside of urban areas. Young unmarried women took this step for a while to provide money for the families they were raised in. After the Civil War, immigrants began to take those jobs. In farm areas, people earned their keep by laboring on their family farm, then moved on to find land of their own farther west when they came of age. Often in these land offerings, there was a requirement of citizenship which excluded Blacks and Native Americans. The economy in this country was 80% agrarian by the end of the 19th century, so any idea that slaves could just "get a job" if freed is misguided. Thus those who wished to abolish slavery proposed land offering to freed slaves (undone by the failure of Reconstruction) or the imaginary "repatriation" to Africa. There was no happy solution as it turned out.
Urban "sweat shops" in New York City originally referred to garment workers crowded into small apartments doing piece work that was distributed to them by their supervisors employed by their employer. It was hard work in a cut-throat business, but it did employ women who had little opportunity to earn at that time except as domestic servants. The city garment factory arose for the most part after the Civil War and the famous Triangle Shirtwaist Fire which killed a large number of young female employees because they were locked in and couldn't get out of the blazing building occurred in 1911. Reaction to the fire caused many reforms in legislated working conditions and fueled the Union movement, led by immigrants who had come to New York toward the end of the 19th Century.
Re slavery, whoever thought that a slave woman who was sold could "bring" her children with her is missing the point of slavery. Those "children" were assets worth money at the slave market and would be themselves sold and the family broken apart. Slavery recognized no family ties, not marriage, not parentage, not siblings. Think of the slaves as pets or cattle, no problem selling any of them off to whatever conditions, no matter how far away. People today do not understand that female slaves were sex slaves at the whim of their master, which explains the 26% Caucasian DNA in today's Black population with roots in slavery all those. "Mulattoes" in the censuses. Evidently until recent years, no educator wanted to stand up in front of a room full of students to paint this picture. From a woman's point of vew, I would rather starve than have to endure that or have to see my daughter endure that. If you listen to music of slavery, a many songs are prayers for death and deliverance to Heaven.
Also the expression "slaves selling themselves"? Who got the money? This is new to me, I would love to see examples. Slaves who were family members of free Blacks were often purchased by their free Black kin to protect them from sale to outsiders by their master, this was the usual situation for free Blacks who were slave owners (but not all of them). And slaves sometimes were capable of working outside of their owner's home to purchase their freedom, this with their owner's consent of course.
There is no comparison between slavery and free factory labor whatsoever, except that injustice and abuse could occur in both. But personal freedom made all the difference.