"Why should the state restrict the church's ability to do so?"
The unspoken arrangement is this. A church is given the privilege of not paying taxes because authoritarian governments, that might not like a particular church, might increase their taxes.
On the other hand, churches are prohibited from getting involved in politics because authoritarian churches might unduly influence the government to their liking. They can say whatever they want in their churches but organizing campaigns in the public is not allowed.
Until Mike Pence came along (Trump's puppeteer) this arrangement worked well.