I think the approach I'd recommend is similar to yours, but perhaps my reasoning will be helpful to share.
In general, we have the same problems with new-user answers on off-topic questions as we do on on-topic ones. They come in three types:
- Not an answer
- Doesn't address the question
- Doesn't attempt to represent the viewpoint requested
- "Thank you" only
- "Bad" answer. Addresses the question, but:
- Strays off the subject
- Reads like a rant
- Is difficult to read due to formatting or grammar
- Gives no sources
- Is simply wrong
- Good answer
Generally speaking, we tend to quickly delete the first group (NAA) but not the second group (though these too can face deletion in the long run if they are not fixed).
Normally, it is fairly easy to evaluate which bucket an answer falls into, particularly when the question is clear and focused. The challenge arises when the question is broad, opinion-based, or otherwise veering into "off-topic" territory. Then it can become difficult to distinguish between "NAA" and "Bad." In such cases, I suggest:
- Vote to close the question (or flag it, if you have less than 3,000 rep)
- Comment on the answer, saying that while we appreciate the effort, the question may be closed due to its off-topic nature
- Take appropriate action on the answer itself.
- If it includes a good-faith attempt to directly answer the question as written (even though it's too broad/opinion-based!), it's not NAA. Downvote and/or edit, or even manually flag for a "citation needed" post notice, but don't flag expecting that a moderator is going to immediately delete it.
- If it doesn't include a good-faith attempt to directly answer the question as written, go ahead and flag for deletion (NAA or VLQ).
Thus, according to this approach, an answer saying "this is what my tradition teaches" will usually be NAA (and immediately deleteable) if it is posted on an on-topic question. But it will not be NAA if the question does not specify which tradition's viewpoint is desired. In that case, it may still be deleted eventually, especially if the question itself is edited or deleted.
Also important: Don't skip the first two steps! If off-topic questions don't get closed, then they can continue to give new users the wrong idea about how this site works. And if answerers aren't told that the question they've answered isn't a good fit, they won't understand why their similar answers to on-topic questions are deleted.