There's no written policy (until now, I guess), and each mod has his or her quirks (so the discussion below is guaranteed to apply only if the handling moderator is yours truly). While you asked about "not constructive", let me be more ambitious and answer in the greatest generality. Before saying anything else, let me first say this:
Moderation on each site is different
... and reflects the evolving culture of the community that uses that website. What applies to StackOverflow does not necessarily apply to Math.SE. And there were many moderation decisions made 3 years ago that, nowadays, probably would have turned out differently.
For a new user who is familiar with other websites in the StackExchange family, a first look at the moderation on this website may give the feeling that the moderation is very lax. Indeed, compare to some of the other more tightly-run SE sites, a lot of what happens on MSE may be described as organised anarchy. Some indications:
- We are generally a lot more permissive about what constitutes on-topic questions here than elsewhere.
- Meta often ending in a bit of a free-for-all.
- Users constantly bickering with each other about "problem statement questions" in the comments.
- And of course, the persistence of comments. (See below; the only other site I know with even higher reluctance to remove comments is MathOverflow. Yes, it has something to do with mathematicians and/or academics.)
Why do we clean up comments?
Comments are by design ephemeral: their edit history is not open to anyone but the developers, and deleted comments can only be seen by diamond moderators. One may argue that the SE philosophy is such that cleaning up of comments (through their deletion and flagging) is encouraged. But we don't believe that comments should be deleted for the sake of deleting comments.
The general (there are some local variations) rule of thumb applied on the main website (by myself and many other mods; meta works somewhat differently) is that:
We delete a comment if and only if its presence does more harm than good.
We do not delete a comment if its presence does neither harm nor good.
In mathematical language:
For "non-constructive", we treat the term "constructive" as a closed condition ("in the French sense").
In what cases would the presence of a comment do harm?
- The comment can be hate speech or just offensive.
- The comment can be commercial spam.
- The presence of a comment can be misleading or confusing to the readers. (For example, comments answering a question which is only presented in a now-deleted comment; comments about a mistake in a previous version of a post.)
- The presence of comments can lead to off-topic discussions. (Keep meta discussions to meta, please.)
- Certain comments, their presence in themselves doing neither good nor harm, can be harmful in the larger scheme of things by making it harder to follow other, actually useful, comment discussions.
- Extremely long comment threads can outlive their usefulness; especially the case where a third party reader will not glean anything useful by unfolding the comments and following the whole discussion.
- Comment threads can become heated, which may lead to full-out flame-wars and overall unpleasantness.
This list is by no means complete, but should give you an idea what moderator(s) look for when they decide whether to act on a comment flag.
Why were your flags declined?
While I wasn't the handling moderator, I can easily guess the reasons.
- A "thank you" comment from the OP is basic human politeness. In the case where the existing comment thread is not very long, its presence basically does no harm.
- The degree of chattiness that each moderator tolerates is different. In my opinion a couple comments by two users familiar with each other about each other's answers is hardly distracting or harmful when they are not obscuring "more important things".
- While downvoters are not required to explain their downvotes, the standing policy of this site has always been that users may ask about the downvotes. They are just expected not to complain if they don't receive an answer.