It wasn't just "choosing not to participate." It was activism seeking to exclude religion from public life. The most important of those being the elimination of religion from public schools in the 1960s. (I should caveat the following by pointing out that the below is not an attempt to convince anybody about what policy would be better. Instead, it's an attempt to describe what I see as a fundamental tension in the approach that we have, for better or worse, adopted.)
A key function of most religions is providing people with a specific moral framework. Fundamental to that is almost always the process of socializing children in that moral framework. (Indeed, the UN recognizes that as a fundamental human right.) At the same time, a key function of schools also is socializing children in the values and morals of their community. For most of American history, there was no conflict between these two institutions--America's overwhelmingly Christian communities socialized their children in their religion through public schools.
The Warren Court's decisions in the 1960s broke that process. Parents had to socialize children in their religion "on their own time." Meanwhile, because teaching morality is an indispensable aspect of socializing children, public schools created an alternative, secular moral framework to impart onto children. Under those precedents, even in communities that were overwhelmingly Christian, and where Christian parents wanted their children raised with a Christian moral framework, public schools were required to teach them something else, in competition with whatever religious instruction they might receive on their own time. It's a unique scenario where attempts to vindicate the individual right of being free from unwanted exposure to religion in schools, has the side effect of dramatically curtailing the ability of the majority in the larger community to impose their religious values into their children.
Taxing people, then using that money to create public schools, and then requiring people to send their kids to those schools, and then prohibiting those schools from teaching kids the religion of their parents, very much has the effect of undermining the propagation of religion. It gives the secular teachings of teachers primacy over whatever religious teachings parents may be able to impart during the waking hours when children aren't in school.
Whether this that approach is nonetheless necessary in a multi-cultural society is a different question. I would point out that countries like Sweden seem to be doing fine allowing religious parents to choose to send their kids to publicly funded religious schools. But I think the effect of promoting secularism and undermining religion is hard to deny.