The first Federal provision for public education in the US dates to 1785, when section 16 of each township was reserved for public school maintenance. So 'government schools', as you put it, predate even our constitution.
So I would say that defending public schools isn't secularist or evolutionist, it's conservative. And people who want to abolish such a basic and traditional part of American society are radicals.
They are huge defenders of government schools, and more significantly, of federal control of education as opposed to state and local control.
I really think the vast majority of parents would be satisfied on the evolution issue if the ACLU and the Politically Correct science community simply practiced what they preach regarding science's stance on religion. They repeat over and over that science is neutral on religion, and can say nothing one way or the other about God's existence.
Yet, what would happen if a few minutes were set aside at the start of every semester to discuss whether a deity created the universe, its laws, and life or whether those things just happen to exist and work the way they do by happenstance? We'd be told that such a discussion would be "unscientific", even though discussions of all kinds of wildly speculative things (alien visitations, parallel universes, etc.) would be considered okay.
Various politically active science organizations would declare the discussion to be a "war against science", they'd check the link list on their websites, phone the ACLU and People for the American Way, and call for a federal judge to ban the discussion.
To them, religious neutrality means operating on the assumption that God doesn't exist. To tell the kiddies that science is ignorant on this issue, and that God may exist or may not, is the same as telling them God exists in their minds.
I think most parents don't want any particular ID theory taught. They just want an acknowledgement that it isn't proven or even provable that we are the descendants of micro-organisms, and that science is as ignorant about the existence of God as Bill Clinton is about morality. But the hardcore evos won't allow such acknowledgements to be given in a public school science class.
In a sense, it doesn't make that much difference any more. More and more parents seem to be opting out of the public schools, and the internet provides ways around public school dogma on evolution. Freer flow of ideas are doing to evolution what talk radio & the internet have done to the liberal media monopoly.