In a previous post, I wrote about a psychological consequence of the separation of church and state: if church and state are separated entirely, this creates a rift in church members' thinking about the state, for they cannot refer to their religious beliefs when thinking about the state. This is impossible, especially regarding moral issues involving the state (e.g., abortion laws, drug policies, military decisions), for moral beliefs belong to religious beliefs.
Consider some sentiments from left-wing Christians. Brenda Davies from the God is Grey YouTube channel says in this Jubilee discussion that "universal healthcare is the most Christ-like thing I can imagine." Exodus 23:9 is also often repeated by left-wing Christians with regard to immigration policy, so it's not surprising that social-justice magazine Sojourners includes it in their article "22 Bible Verses on Welcoming Immigrants."
Perhaps I'm being speculative here, but I believe there's a certain kind of person who claims to endorse the separation of church and state yet has no problem if religion is used to inform certain laws and policies, so long as religion informs those laws and policies in their preferred way. This group I am imagining would of course include progressive Christians, but it could also include non-religious progressives who didn't mind getting the political ends they wanted by religious means.
What does the separation of church and state apply to, then? It seems to me that it applies mostly to matters of a sexual nature, such as divorce, gay marriage, abortion, and trans rights. This is selective and inconsistent.
As far as I can tell, "the separation of church and state" is usually shorthand for not applying certain ethical standards of a religious community to those outside who would find those ethical standards unpleasant.
So it's not really about separating the church and the state at all. It's about removing any obstacles that might get in the way of one's preferred laws being commenced or policies being enacted.