> The first part just means that Congress cannot regulate established state churches

Err, no, that's a common misunderstanding.

The Establishment Clause of the First Amendment prohibits government from encouraging or promoting ("establishing") religion in any way.

It's there to forbid the creation of an official state religion (eg: the Church of England being created to replace the Catholic church back in the day).

See:

https://www.aclu.org/other/your-right-religious-freedom