With regards to your first question, it's a bit of a chicken and egg situation. It was less that American decided to not fund religious schools and more that aspects of Protestantism^1 were so embedded in early American culture, common schools were basically religious schools. This isn't to say they explicitly taught religion, though some did, but it was in everything from notions of good behavior to texts in primers and readers. Protestant sects that were outside the main faith, most notably Quakers, opened their own schools. Unlike the conservative religious schools that were to come in the late 19th and 20th century, Quaker parents weren't necessarily avoiding public school content, it was more their religious traditions were such they were seeing to their children's education in their own way made better sense financially and socially.

It's also worth a few sentences to talk about the issue of scale and size in American education history. First, American education is locally controlled. It wasn't the mid-20th century that the federal government truly had any involvement in day to day operations of schools. This meant each state determined how education was to work. And the nature of state-level involvement in local schools was highly varied, especially after the Civil War. Second, a large part of the rise of public schools in the parts of America outside the east coast involved young, unmarried white women (and some men) heading west after a short teacher induction program. They were recruited to run schools in the settlements, forts, towns, and villages across the country and were almost always the only educator around. That is, it wouldn't be until the advent of the modern school transportation system that an area would have more than one public school. So, taxes went to support only one school building. Urban areas were, of course, different and every city in the country has its own education history.

However, Sunday (or Saturday) school as supplemental education for a community's children was fairly common. A community's children might attend the local common school and receive a basic, foundational Protestantism-infused, liberal arts education for a few hours a day Monday through Friday for a few weeks every few months and spend an occasional Sunday mornings receiving explicit religious instruction. Generally speaking, communities saw the schools as serving two different purposes - one was about a child's cognitive development, one was about their spiritual development. Basically, if a community had social norms around literacy, children went to the common school. If the social norms involved religious study, they'd also go to Sunday school. (Unless their religious traditions were such mainstream Protestantism was in contradiction to their faith, then were educated at home or attended a school run by their community of faith.)

Which is to say, in many cases, there simply weren't enough small bodies to justify or rationalize opening a second school focused on different religious traditions. Until there were. Which is what happened in New York City between Catholics and Protestants beginning around 1830 or so and continuing into the modern era. The best history of the period is by D. Ravitch and called The Great School Wars (2000). (Amusingly, I discovered a British author was inspired by the title for a book about Britain's schools) "War" is a fairly common descriptor in books on education history (The Teacher Wars, Testing Wars, etc.) and usually it's a bit of poetic license. When it comes to what happened in NYC, it's a fairly fitting description.

One of the sticking points for a number of decades in the 1800's, was convincing men of means to contribute to the education of other men's children. (In the South, that meant convincing wealthy white men to contribute to a fund that might, possibly go to educating Black children and is its own entire history. Hashtag local control.) Convincing them took a lot of speeches and persuasion around the benefits of an educated population for the country. Advocates leaned into the notion of Republican motherhood as an argument for educating daughters as future mothers of voters and that even a poor man's son could be taught to be a good American. Most of that pushback died down by the end of the 19th century as school became more common and high schools (often described as "the people's college") spread across the country.

A great deal of that "other men's sons" animosity rose back up with regards to educating Catholic children but it took a different form. Xenophobia was the norm and the Protestant men running NYC's public schools were fine with Catholic children in their schools - in fact, they encouraged it, firm in their conviction children would walk in Russian, Irish Catholic, Italian Catholic, etc. and walk out American. They felt American schools could educate the Catholic right out of them - but Catholic parents were deeply troubled by the strands of Protestantism that were weaved into the cloth of American public schools. Catholic clergy built Catholic, tuition-free schools but they simply couldn't keep up with the numbers. In 1820, members of a Baptist Church in NYC built a school and applied for public dollars to run the property, which angered those who ran the public schools. In 1825, they passed an ordinance that no common school dollars could go to a school operated by a church of any denomination (but again, keep in mind public schools were Protestant-lite, complete with religious parables in readers.)

I won't summarize the entire book (it's really good! There are Irish Catholic priests who got into brawls! Mothers breaking into schools to raise havoc! Boss Tweed!), but suffice to say when the wars ended, NYC had two parallel school systems - the public schools and the diocesan schools. Eventually, a network of yeshivas and madrassas were established. Not the mention the Quaker Friends schools and an entire, insular network of tony secular private schools. There's a whole history behind the modern system of public funding used towards private schools and to be sure, it's a mess. Likewise, there's an entire legal history tied up in the secularization of American schools in the 20th century or as it's colloquially referred to as, "taking God out of schools."

To your second question, I have to defer to those better versed in the history of Britain's educational history. My understanding, though, from this side of the pond, is that generally speaking, it can be attributed to the centralized nature of the British education and the role of population centers. The exact history of it is something I look forward to learning!

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1. The term WASP - White/Wealthy Anglo-Saxon Protestant - emerged as a way to describe the culture of the men who set the parameters around a great deal of how America functioned, especially in New England.