Was the U.S. the first country with Public Schooling?

The short answer is "no" and the longer answer is "it depends on how we define "first", "all", and "traditional."

Regarding "traditional", governments throughout history have had varying degrees of interest, oversight, and influence on the education of a country's children (which has mostly meant the sons of men with social status.) For example, the Persian and Spartan states provided an education to boys from the age of seven until 15. The Athens government built and paid for the upkeep for gymnasia for educational purposes. While the academies were privately funded, the state had a say over class size, teacher qualifications, and location. Finally, the Jixia Academy located in the Qi state of China was founded around 300 BCE by King Xuan with the goal of providing a state-funded education to interested scholars.^1 While children through various ages and in various places received educations from their parents, tutors, at church-sponsored schools or academies their parents funded, it's hard to say there's a universal tradition of education being a private endeavor.

Generally speaking, we can see the evolution of a public education system for all children if we look at compulsory education laws, which, unless otherwise specified, apply to children of all genders, ethnicities, social status, and religion within the state. America provides an example of the complexities of this. By the end of 1920's, states, including Mississippi, had laws on the books requiring public schools provide an education to every child that showed up at a public school. However, the state repealed their laws around the time of Brown v. Board of Education of Topeka. The repeal was clearly about keeping Black children out of schools funded with tax dollars. It meant that if a Black parent tried to enroll their child in a particular school, the school could refuse the child admission, claiming they had no obligation to provide the child an education. The law wouldn't go back on the books until 1972. Additionally, prior to the passage of the Individuals with Disabilities Education Act (IDEA) in 1975, it wasn't uncommon for schools or districts to refuse admission to children with disabilities they deemed too severe for inclusion in public education.

"First" gets into some murky territory in terms of nation-states and open access to all children. Prussia is generally recognized as the first European country to institutionalize public education for children of all genders and from all social classes in 1763. By 1870, 67% of school-aged children were enrolled in a state-funded school. Having laws, though, didn't necessarily mean all families would chose public education. Greece passed compulsory education laws in 1834 and by 1870, only 20% of children were enrolled.^2 Prussia, though, ceased to be Prussia before cultural norms shifted around the education of children with disabilities so it's hard to say they provided an education to all children. And while some regions of the US colonies had ad hoc systems of private and charity schools funded with land grants and tax dollars prior to 1776, they were were small and regional and typically excluded white girls, Black or Indigenous children, and white boys with disabilities. So, no, the United States wasn't first.

Hartnett, R. A. (2011). The Jixia Academy and the birth of higher learning in China: A comparison of fourth-century BC Chinese education with ancient Greece. Edwin Mellen Press.
Soysal, Y. N., & Strang, D. (1989). Construction of the first mass education systems in nineteenth-century Europe. Sociology of education, 277-288.