Great question! Thanks for asking!

The challenge is that an estimate is the best you're going to arrive at given how idiosyncratic education was in that era. There weren't (and still aren't) any levers or structures for ensuring things happen at a certain time in a certain order in American schools.

David Page, who mentioned handwriting instruction in his writing, actually provides an interesting example of what that means; he was a New York State educator. His writing on education was intended for New York educators, in particular those who attended New York Normal (teacher prep) programs. This isn't to say he wasn't thinking about teachers in other states, but rather, he was writing from a NY state perspective, where public education had been part of the conversation since the late 1700s. He didn't have to persuade those reading his writing that teaching children cursive was a good idea, he knew his readers were already on board for the idea. In contrast, Illinois wasn't a state until 1818. Educators in NYS had a generation or two to think about what should be happening in schools before "Illinois school children" were even a thing.

In other words, it's difficult to take the writing of any educator from the 1800s and apply it nationally. To put it another way, at the time when Page was writing, there were still thousands (if not millions) of Black and African enslaved children. There was no formal structure for their education and in the limited cases where they were being taught, the timing came down to circumstance and need. If an enslaver wanted/needed a child to be literate and able to write, it would happen on a timeline that worked for the enslaver. (Frederick Douglass provides an example of this in his autobiography in Chapter VII.) Meanwhile, the white son of an enslaver might be working with a tutor as soon as his parents thought him ready and could have mastered cursive writing by the time he was 8 or 9 years old.

The idiosyncratic nature of education is also some schools in Illinois were called "ABC schools." It's was a colloquial term used in the state to describe a particular kind of free school that appears to be only used in the one state. From an 1857 Illinois paper:

Our school directors having exhausted their resources in erecting two large building, and organizing them in very superior schools, the people of the other districts in the city are unwilling to send their children to the inferior A B C schools provided for them, and a general rush from all quarters is made for the large schools.

In contrast, such schools were sometimes called "academies" in New England states. Even the names of different kinds of schools were varied based on region in that era.

It's also helpful to contextualize what was happening in the larger picture of American education. The feminization of the profession was mostly complete by the era you're asking about; teaching was woman's work (mostly), leadership or administration was men's work (mostly). These men often referred to as "schoolmen," were expected to do a great deal of writing. Lots and lots of writing. And pontificating. So, in texts of the era, it's not uncommon to find proclamations about what should or does happen in school but it's difficult to say their statements matched what was actually happening in the classroom. At the same time, authors of the era - especially those in education working to change minds and policy - often had a bit of a heavy rhetorical hand. For example, it's not uncommon to find letters where teachers talk about children going blind from reading in schoolhouses with poor lighting, or teachers writing about heading for an early grave because of bad the ventilation is in the school, and oh by the way, could you please send more money so we can build a better schoolhouse?

Which is to say, if you do find a text from the era that makes a proclamation about when handwriting is being taught, especially one that claims to speak for teachers beyond their particular region or state, especially one that bases its argument on the human body, it should be taken with a heavy grain of salt.

This doesn't mean, though, historians can't speak to larger patterns. This 1917 piece on the history of teaching handwriting provides some interesting, larger patterns and Tamara Plakins Thornton's Handwriting in America: A Cultural History is a fascinating read.

All of this means it's difficult to confidently say, "In the 1870s, Illinois children started learning cursive when they were 9 years old." Instead, it's more of a, "based on their race, gender, geographical location, family financial situation, and the philosophical and pedagogical constructs of the adults around them, Illinois children enrolled in school in 1860s/70s may have started learning cursive after demonstrating mastery of their print letters, or they may have learned cursive in conjunction with print. It depends."