American religions are supposed to stay out of politics, or they risk their tax-exempt status.
For me, the disturbing event was shortly before the 2016 event when a Catholic Church in Lowell MA had posters urging people to vote no on marijuana legalization.
(In my case, I smelt the politization when I was a teenager so I never continued being Catholic as an adult.)
Unfortunately, those laws are not currently enforced.
I see another reply is arguing that religion is inherently political. I disagree. Modern politics did not exist at the time the world's major religions were being formed. Attempting to twist them to fit with a particular party or candidate is a terrible idea all around in my book, for many reasons.
The state can cause a lot of damage by endorsing religion, the historical record is overflowing with examples. I'd argue that a religious body endorsing a state is every bit as potentially destructive. The government is at its best when it is neutral on the subject of faith and crafts policy in evidence-based ways that people of all (or at least most) faiths can agree upon. In this situation, there is no reason for a religious organization to promote such a government because their interests are orthogonal. They can cooperate, but there is a clear line between that and acting subservient, or declaring some sort of 'divine mandate' has been bequeathed upon a government institution or official.
American religions are more like American Indian tribal nations. They have independent jurisdiction and their income is not subject to taxation. Whether or not they engage in politics is completely their prerogative and has no bearing on their tax exempt status. It’s like saying the Navajo nation can’t engage in politics or else they would lose their tax exemption.
Further, the core reason for freedom of speech in a democracy is to have freedom for political speech. The need is to have different factions discuss ideas related to the governing of society. Any legal regime that restricts the rights of religion to engage in political speech is one that rejects the separation of church and state. The purpose of the separation is to prevent the government from interfering with the rights of disfavored religious groups or granting special privileges to favored religions. If an individual has a right to political speech, then an association of individuals also has that right whether or not it is religious in nature.
When my state was debating creating a state-run lottery to fund education projects, my preacher gave a sermon on the evils of gambling. Religions can't realistically stay out of politics because every law can be reinterpreted as a moral argument.
This is incorrect.
(1) Religions are treated no differently than any other non-profit.
(2) No non-profit may endorse a particular candidate. They are free to comment on particular issues and policies and referenda.
A priest can say 'vote no on marijuana'. They cannot say 'vote yes to Mary Sue because she doesn't like marijuana'.
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.