It is about subtlety if the point is to examine and subvert a tired trope, which they did. When the fanbase whines about being the Chosen One when you're the opposite of a Chosen One, then it's time to reexamine the text, so to speak.
I would say rather than being a good examination of the role of organized religion in a setting, DAI is excellent at personal faith and self-identity. Leliana and Cassandra are good examples of this, as is the Inquisitor and spirit!Justinia in the Fade. There is no point where our realization of how we got the Anchor affects how the rest of Thedas sees us, we will still be treated as the Herald for the rest of the game.
Trespasser does tarnish our image, but that's more of a political reason than religious. One of our options is to remain as the Divine's personal guard, so the Inquisitor remains an influential figure in the Chantry.
I don't think the tired trope was really subverted that much. We find out we're not Andraste's Herald. We face inner doubts about our role now that we know we're not chosen. The game continues on with no visceral reaction from the general populace/Chantry about this revelation. Most likely, our advisors don't even allow us to make this public knowledge.
The only possible changes to organized religion in Thedas are due to the choice of Divine, not anything to do with the Chosen One PC. There's no real discussion about schisms in the populace divided between those who believe we're the Herald and those who don't. This should have brought about a Reformation-style rupture within the Chantry or at least discussed the implications of this, but even after we return from the Fade everyone still treats us as the Herald.
The game is very good in highlighting personal faith and belief, but I wouldn't say it's great on organized religion.
Also, I wouldn't say the PC is the opposite of a Chosen One as much as a variation of a Chosen One trope. We're still the only person with a unique ability that is required to save the world. Just because we received that ability by pure chance rather than divine intervention doesn't negate the fact that we're still the only one with the world saving ability. We're the Chosen One of blind luck instead of the Maker.