The "Christian Evangelical right" has also been pushing school systems all across the country for years to eliminate any educational premise that doesn't conform to the "Word of God", e.g.- Darwin's evolutional theories. So, they are trying to tell you what to teach your children and what to expose your children to.
My specific example: In Oklahoma in the late 1970s when I was a child at school, the Southern Baptist Convention was pressing hard for schools in many states to focus on teachings aligned with religiously-based foundations for creationism, rather than scientific evolutionary proof. I grew up in the "Bible Belt" of the south, so I know what I'm talking about when it comes to this.
And, it is still happening to this day across the country. Prayer in school, religious depictions of holidays, etc. But, wanting to exclude religious practices other than those of Christian faiths, e.g.- in my hometown, there were Jewish kids in my school, yet we never learned about Jewish holidays in school...only the Christian version of Christmas and Jesus.
They also were trying to dictate morality in telling school systems where I lived by saying that it was wrong to have school dances because, according to the Southern Baptist Convention, it was an abomination against the Lord unless you were dancing in His "Glory".
And in fact, the First Baptist Church in my hometown single-handedly managed to cause a Kroger store to close (through patron intimidation and members of their church being on every city board) because they were selling beer in a grocery store and that was "immoral" according to their pastor. So, there's your religious right interfering in business as well and telling people how to live (legal adults couldn't get beer at a retail grocery establishment because of religious influence).
Hence, the "Christian Evangelical right" (whom aligns themselves so closely to the Republican party and to whom the Republican party caters to most for votes) is trying to dictate morality in schools, homes, and workplaces. Hence, their effect is going to be felt most through Republican-led initiatives within society, since it is the Republicans who are expected to "pay back" those religious campaign supporters via initiatives in legislation that give power to their religious causes.
As for "school choice" with vouchers...the right doesn't want "to have true choice and competition in public schools". They want school vouchers to help pay for them to send their kids to the private school of their choice. Plain and simple.
And let's not get confused here:
Private schools are NOT public schools. And if you are not happy with the education your children receive in the public system, you have a right to CHOOSE to send them to a private school at your cost which will teach them what you find best. And, the right WAS pushing lawmakers to make it legal for school vouchers to be used to fund private educations. That was happening in Florida where I live.
Anyways...the church tries to interfere and they're not absolved simply because they consider themselves of the highest moral calibre.
So with my examples, they (the Christian right) are guilty of trying to tell people what their kids should learn, what extracirricular activities their children should attend, where they should spend their money, and on what they should spend it...all based on their own faith.
And for they in the religious right to assert their religious beliefs on others, through intimidation or legal code, is a violation of those other peoples' right to freedom of religion.
