The reasoning behind the religious aspect of 1st amendment was because the people who paid taxes to the state didn't want their tax dollars benefiting a faith they were not part of. What I find funny is that Christians are wanting to allow various religious practices in the schools. What they fail to realize is that other religions would have the same rights. What do you think would happen if a child came home and told mom and dad "I heard a lot about Wiccan in school today and want to join them." How long do you think it would take for the parents to complain about religion being part of school then? Hey, if you open a door, you must leave it open for the rest. Is that a door they want to open?
Of course, the first religion people think about whenever they hear "radical" would be Islam, due to the media portraying most Muslims as terrorists in the modern world. However, this idea would apply just as well towards the politics of the West, as plenty of times we see people (more prominently in the U.S.) do things because of their views that do not make sense for the well-being of the nation.
Two examples in the Middle East would be Saudi Arabia and Israel/Palestine.
In SA, religion and the monarchy dominate politics, and that's why people call it 'ultra-conservative'. They were not willing to reform or change anything due to their views forbidding them to do so. Only in the last few years has the monarchy's grip on religious politics softened - the first movie theatre was opened recently and women are now allowed to drive.
Meanwhile, the Israel/Palestine conflict is possibly the biggest example I can think of here because the main reason the conflict existed, to begin with, has to be the religious views of the Israeli Jews and the Palestinian Muslims, as their views conflict when they claim their homeland, leading to the conflict we see today, the same as it was in 1948 when it first arose. If church did not dominate politics, the conflict would be solved, likely in a one-state, unified Israeli-Palestine.
The West, however, also sees its religion dominate politics, even today. For example, about 43% of Americans would not vote for an atheist to be their president. Why? Likely because they fear that what they do would conflict with their beliefs, which is what has caused a lot of political upheaval in the U.S. especially. With such a religiously diverse nation, there of course come the moments where people make headlines for the Supreme Court case of "religious person did this to someone". Take the wedding cake case from 2017 - link will be below.
I do understand that since many people are religious, they would not favour separation of church and state. This would likely be the case for a country like, say, Argentina, where almost 9 out of 10 people are Christian. But in a place like the U.S., where there are so many religions intertwined (much more than in the past, at least), I believe that the separation of the religion and the state would overall lead to a happier populace.
Atheists for president - http://news.gallup.com/poll/155285/atheists-muslims-bias-presidential-candidates.aspx
Gay wedding cake case - https://www.cnn.com/2017/12/05/politics/supreme-court-masterpiece-cakeshop/index.html
Argentinian Christian population - https://en.wikipedia.org/wiki/Christianity_by_country