You can't conquer by the sword if you don't yet have a sword. Once the Church got the sword (imperial power), it conquered and ruled by it when necessary.

No, indeed the Christian narrative was sometimes popular. There's no accounting for taste, after all. But even when populations converted relatively peacefully, the Church maintained order (thought control) by force under the pretext that it was battling demons and ushering in the imminent end of the world.

The Church attempted to conquer the New World largely by force. It killed plenty of Indigenous Canadian children in "schools" or forced conversion camps, for instance, as the recently discovered mass graves reveal.