It's straight up culture war against Christian beliefs of any kind, but especially moral beliefs concerning sex.
More broadly, any cultural or moral beliefs, or indeed any Weltanschaungen or philosophical views, that seriously oppose the convictions of the liberal elites, their PC morality or their incoherent, secularist, half-atheist half-scientism, are the enemy.
Since the mid-20th Century, federal courts and federal bureaucracies have nationalized our 50 state systems of public education, converting them into increasingly homogenized machines for raising kids with liberal-approved attitudes, habits, customs, ambitions, and beliefs.
Parents have since that same time sought the ability to send their kids to private and especially church-affiliated schools escaping liberal secularist/atheist ideological control as well as liberal racial/ethnic mixing and balancing with at least public aid if not complete public finance.
Just as they used separationism and the federal courts to secularize the public schools, liberals have attempted to use them to disallow school voucher plans, but the courts have so far held that vouchers, like the GI Bill, are assistance to parents or students and not constitutionally impermissible payments to religious organizations.
At the same time, however, liberals have sought to get the courts to impose a degree of compliance with liberal PC ideology on schools getting public funds, specifically regarding science education (global warming and evolution) and racial, ethnic, and moral (LGBT) nondiscrimination in staffing and admissions, as well as in the manner and content of their teachings (forcing homosexuality off the sin list, as Frank Bruni describes it).
So far, that hasn't worked, either.
Too, liberals are unhappy church-affiliated schools can require ideological and moral acceptance of and compliance with church doctrine from staff, though they expect public schools to impose analogous, liberal requirements on their staff.
None of this is necessarily connected to class politics, and both popular Protestantism and Catholicism were allied with progressivism from the beginning until the merger of the latter with social and cultural liberalism, and the attacks of the latter on American clericalism from about the 1950s.