The Society of Friends (AKA Quakers) doesn't have professional clergy, and decisions are made democratically. They don't even have pastors; my parents were married in a Quaker meeting house, and their marriage license was signed by the secretary of the congregation.

Here's an article about their decision-making process: http://quaker.org/legacy/tqe/2005/TQE118-EN-Democracy.html

Most Protestant churches are run democratically with voting on elders and so forth

It isn't clear that this leads to "better" outcomes, though. The larger denominations are pretty standardized, the smaller churches with more independence are where you find more sectarianism, extremism, and even cult-like behavior.

That's a good point. When it comes to 'how to run a church' it is a pretty good basis, unfortunately protestantism is very fragmented, but then again, if it wasn't then it would be more like the structure it broke away from so maybe that is actually a good thing.

> Have there ever been democratic or republic churches?

Yes, to the point where there's vocabulary to describe those types of organization, e.g.:

https://en.wikipedia.org/wiki/Presbyterian_polity

https://en.wikipedia.org/wiki/Congregationalist_polity

I can't speak to democratic/republic, but a number of syncretic religions and alternative spiritualities are either decentralized or so fragmented as to be functionally so. In my experience, such provide several of the social benefits of religion - i.e. gatherings of like-minded people with shared theologies - with reduced potential for abuse of power.

Of course, the scalability of these systems are unknown.