“Separation of church and state” doesn’t require that at all. It’s required by churches’ non-profit status, the same as non-religious non-profits. But non-religious non-profits also get involved in political speech in every election.
This is the same thing as where people fail to distinguish the First Amendment from free speech as a principle.
If we are talking about law, you are right - the US limits on religious groups getting involved in politics are in the tax code, not in the establishment clause. If they are willing to give up their taxation privileges, they are allowed to participate in politics as much as they like. And, since the tax code is just a statute, Congress could change it to allow them to meddle in politics without losing those tax privileges, and I doubt such a tax law change would be held to violate the establishment clause.
However, “separation of church and state” is also an extralegal principle. And, maybe, too much involvement of religions in politics violates that extralegal principle even when it doesn’t violate the letter of the law.
