As with most events involving children in American history, there are a variety of contributing factors that influence their place in society and how adults seem them. The end of the 19th century, beginning of the 20th, marked the convergence of a number of factors that led to the decline in child labor. These include the spread of pediatrics as a separate medical field, the identification of "adolescence" as a distinct phase in life, changes in census documentation and data collection, tensions between unions, laborers, and manufactures around who exactly should be doing the work of labor, and finally, my bailiwick, the rise of compulsory public education and high school. They all played different roles and exerted pressure on the system in different ways and at different times.

I'll defer to others regarding the non-education variables, but can speak to how compulsory education impacted the child labor situation. The first push against child labor came with the end of the Civil War and the emancipation of enslaved people. Prior to the war, children worked in the fields alongside their parents, provided they hadn't been sold away to another farm or plantation. Like the enslaved adults around them, they were forbidden from receiving an education. In effect, the presences of thousands of newly emancipated children, and their relatives, looking for an education during Reconstruction put pressure on southern states to establish formal systems of education that went beyond the ad hoc systems that had existed leading up to the Civil War. There would be no universal school experience and some Black children worked on sharecropper farms and some white families maintained a system in house private tutors but Southern states started to navigate toward a system similar to what could be seen in states that remained in the union; a system of tax-payer funded schools that provided a liberal arts curriculum to all* of a communities' children. (*All didn't mean all - Black students were not allowed and/or expected by white parents to enroll in schools attended by white children. White children with identifiable disabilities could find their attendance dependent on the whims and personal bias of parents or those running the school.)

Although there is no national education system (due to the 10th Amendment and the lack of an explicit reference to education in the Constitution), the period from the end of the Civil War to the end of the century marked a norming across states. While most of the New England states, including New York had compulsory education laws on the books since the late 1700's, early 1800's, they began to move from a dead letter status to enforcement. In most cases, the laws required either a town provide a school or parents send them and rarely specified for how long or for what purpose. This norming process meant that the common school movement that started in the 1840's settled into a public school movement wherein parents routinely sent their children from around age 5 or so to age 13 or so (1st to 8th grade.) Meanwhile, the historical "classical" curriculum gave way to the modern liberal arts curriculum. Beginning in the 1880's, states extended funding for high schools, meaning children could attend until they were 16 or 17.

Overwhelmingly, children stopping going to school when they were done with school - and the reason for "doneness" were extensive and diverse. In some cases, it was because parents felt they were done. In others, the child was needed for to care for younger children or they needed to go work. Trying to strengthen the system and create a more effective and efficient bureaucracy meant school leaders and politicians across the country established formal structures. In some cases, the focus was on getting the children of poor white laborers in the classroom next to the sons of wealthy white manufacturers and tradespeople. In others, it was on keeping Black children away from white children. Despite the highly localized nature of education in American, a grammar began to emerge; teaching shifted to a feminized job, the school day and year evened out from a few hours in the morning and a few weeks every few months the entire year to extended breaks in winter and summer (mostly due to when wealthier and then middle class families left the cities for vacation), and a shared curriculum that can be described as Americana.

From a previous response of mine.

"Americana" - the simplified versions about American history that are passed on through popular culture and in American schools that serve to communicate lessons about who or why America was founded; as a country and an idea. These simplified histories are typically centered on the experiences of Europeans or white Americans and focus on traits like courage, honesty, and loyalty.

The roots of Americana can be found in Protestant teachings, which shaped early education curriculum as well as the deeply sexist belief that women teachers could only handle so much complex instruction. Regardless, it took hold. So, once we get to the late 1890's and early 1900's, school is increasingly something children do. The full brunt of social pressure around attending school for 13 years wouldn't hit until the 1950's and 60's but even around 1900, the pressure was there. And it was something children did, not because an education prepared them for a career or higher education, but because school taught children how to be America. Which made school very appealing to immigrant families. From another previous answer related to the immigrant experience:

Immigrants typically decided for themselves how, when, and to what degree they became "American." And it was very time, family, and location-specific. Some parents made the choice to never speak their native language in front of their children to as not impede their Americanization while other parents kept their child home if the school insisted children only speak English. Girls from one tenement building might see their brothers go to school while they were required to work in a factory. Boys in the next district were expected to work alongside their parents and didn't attend school until the local district hired a truancy officer explicitly to track them down. However, as states and school districts moved to a per pupil funding structure, more and more children were moved from the factory to the classroom, even, at times, overriding parents' wishes.

In some cases, most notably for German families, there were enclaves across the east coast and midwest where they would be surrounded by the German language, food, and cultural touchstones. Their children could go to a public school (mostly for free, funded by tax dollars, or a small fee which went towards the teacher(s) salary and school upkeep) and learn English and the lessons of Americana. In many cases, teachers taught stories about Columbus, Washington, and the founding of America in both German and English.There are documented instances of teachers chiding children for speaking German but there were also cases where teachers allowed children to speak whatever language they wanted, as long as they were kind to children who spoke different languages. (Getting along with people from different backgrounds was/is seen as an American trait.) In a real sense, this meant that the immigrants were American from the moment they arrived on American soil.

The surge in Irish immigrants to New York City in the early 1900's led to standoff between the Protestant school leaders and Catholic parents. While Irish Catholic parents were fine sending their children to American schools, they refused to endorse a system where learning to be America meant reading Protestant texts, which featured heavily in primers and texts of the era (alongside stories about Washington, et al) When the battle was over, NYC had two parallel school systems: the public secular and the parochial one. The parochial system, though, maintained many of the non-Protestant American touchstones, especially in history instruction.

So, basically, in 1900, recently arrived immigrants were choosing to enroll their child in school rather than send them to work in other to expedite the child's Americanization AND school leaders were putting pressure on all families to send their children to school. The combination of these pressures contributed to a softening of the ground under the social acceptableness of sending a child to work in a factor or field, contributing to the eventual elimination of the practice.