The "act accordingly" part is where I disagree. Obviously Christians can be immoral. Authoritarian? I'm biased of course, being a Christian anarchist, but Jesus rejection of exerting the force of wordly power is so close to critical and central to his tenants that I feel skepticism ->

The Catholic Church has an authoritarian structure. Arguing that makes it not really Christian does not pass the smell test.

About their label is called for. The same way we can spot improper use of words and their meanings when these people do it in other instances. At any rate I broadly agree with your point.