The following post Enforcement of Quality Standards give directions.
If you encounter a user who consistently answers low-quality (or duplicate) questions over a significant period of time, and polite references to this post do not seem to be helping, please raise a flag on one of their answers.
This means, no you should not flag each and every answer to a question that might deserve closure. You should do this only in the case indicated, that is, flag in case "you encounter a user who consistently answers low-quality (or duplicate) questions over a significant period of time."
The policy is about recurring behavior. Moreover, it is not really about all questions that might need closure, but that is a detail.
To avoid confusion "flags" here refer to "flags" for "in need of moderator intervention". If a particular answer is "not an answer" or the answer post itself is "low quality" the posts can and should be flagged as such (by contrast an answer post should not be flagged as "low quality" if the post itself is not "low quality"). The same goes for "spam" and "rude and abusive" flags but that's likely more clear.
When engaging with users (old or new) who have posted (or answered) a low-quality question, remember to remain polite and courteous. Please do not issue unpleasant orders or leave judgemental comments: we will not tolerate rudeness, even when in the context of a honest attempt to improve the quality of the site.
I hope this clarifies the situation a bit.
I'll note that OP had actually a problem with one of my answers, too, but the other post is rather explicit. Thus, this is not a post hoc justification.