I have to disagree with you on that latter part.
But first, yes monks can convert by just being in a province, using them on "minister to settlement" mission increases province happiness (I think 1 happiness per star)
If you really want high level monks, you should use them aggressively: demoralise enemy armies and incite revolts. As Mori I actually incited so many revolts in Kyushu that I got the island with just 1 war declaration. Better yet you should capture holy site and build pilgrim hostel so you have rank 3 monks right off the bat, only downside being no retainers until level up, which will take longer.
Also religion is not important in this game. There are 3 religions, and Ikko Ikki usually get wiped out early/mid game, and so far I've only had Kyushu clans and Chosokabe convert to Christianity, and they're all so wrapped up on their own islands that they won't exactly spread their religion elsewhere.
I see religious buildings and monks as just one more way to boost my chi research and happiness, not a big fan of warrior monks.
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