The separation of church and state in the US was for the state to stay out of religion.
(the US was founded by religious exiles from a state which didn't stay out)
Religions are explicitly political but politics shouldn't interfere with religions. To follow your religion means interacting with the outside world. It's not some personally private thing like a harmless badge you wear (although there are American faith communities that advocate for that).
The cases in the past where political have interfered with religions are often, ironically enough, by other religious politicians. Hence the good idea to separate church and state.
Religion is a powerful tool for politicians in sheep's clothing, but God's kingdom is "explicitly" (i.e. stated in the Bible to be) not of this earth.
From the horse's mouth:
https://www.irs.gov/newsroom/charities-churches-and-politics
> In 1954, Congress approved an amendment by Sen. Lyndon Johnson to prohibit 501(c)(3) organizations, which includes charities and churches, from engaging in any political campaign activity. To the extent Congress has revisited the ban over the years, it has in fact strengthened the ban. The most recent change came in 1987 when Congress amended the language to clarify that the prohibition also applies to statements opposing candidates.
> Currently, the law prohibits political campaign activity by charities and churches by defining a 501(c)(3) organization as one "which does not participate in, or intervene in (including the publishing or distributing of statements), any political campaign on behalf of (or in opposition to) any candidate for public office."