For questions of extremely low quality that are sure to get closed anyway, do the moderators still want flags? Or is that just cluttering up their mailboxes to no purpose?
It seems that we are mostly talking here about close flags and very low quality flags - since these are the two types of flags which seem reasonable in the case of low quality questions (whichever interpretation of the phrase low quality you choose.)
Since the OPs main concern seems to be whether the flag adds work for moderators, it is worth mentioning that these types of flags in the ideal scenario are actually not handled by mods but by other members of the community. The help on flags explicitly says: "Many flags are handled by other members of the site like yourself, who've earned review privileges. Close flags, Not an Answer, and Very Low Quality are primarily handled this way."
And even the flags which are handled by moderators do not get to their mailboxes, but I assume you were talking figuratively in the question.
When you flag a post as low quality or should be closed, it enters a review queue where other users vote what happens with the post next.
Low quality flag sends a question into the low quality posts review queue. For questions which enter this review queue, 2k+ users can choose from the option Looks OK, Edit, Close (or Recommend Close). For answers the last option is Delete (or Recommend Deletion) instead.
If you choose should be closed flag, the question enters the close votes review queue. In this review queue, 3k+ users can choose between the options Edit, Leave Open, Close. (As already mentioned in the comment, for users with privilege to cast close votes, there is no actual difference between this type of flag and casting a close vote. A flag is automatically converted to a close vote. However, a close vote pushes the question into the review queue, too.)
Since the OP asks also how much work flags add to reviewers (or moderators) I will add that you can get at least some idea from the review history (available to users with sufficient reputation) or from today's stats (which are shown to all users). Here is a link to review history and review stats for close votes review. And here is review history and review stats for low quality posts review.
For the benefit of users who (like me) do not have access to the review history, I will list here the most recent low quality posts and close votes reviews. I hope I did not miss some of them. I covered at least the last 24 hours. (I checked reviews with ids starting with 82871 until now. The most recent one I found was 82950.) It is perhaps not very surprising that in the period I checked I found much more close votes reviews than low quality posts reviews.
- Close vote reviews: 82876, 82877, 82880, 82881, 82883, 82893, 82903, 82905, 82907, 82911, 82912, 82913, 82917, 82919, 82921, 82922, 82925, 82928, 82932, 82937, 82939, 82940, 82942, 82946, 82948, 82950. (26 reviews)
- Low quality post reviews: 82871, 82915, 82931, 82941. (4 reviews)
If you haven't sufficient reputation to look at the review history, you could use SEDE, too. But keep in mind that the data in SEDE is only updated once a week. Here is a query listing all reviews. And you here is a query where you can restrict to some specific types of reviews by changing the parameters.
For details on when a review of a post is finished, see here: When is a review task "completed" or "invalidated"? A question is removed both from the close votes review queue and from the low quality posts review queue immediately after it is closed (or deleted).
Some network-wide guidelines on what posts should be flagged as VLQ can be found here: When to use the flag “low quality”? The official description of the low quality flag is: "This question/answer has severe formatting or content problems. This question/answer is unlikely to be salvageable through editing, and might need to be removed."
I will add that for posts which are bad enough to consider a low quality flag, users with sufficient reputation might also consider voting to delete. (The reputation needed is 10k for questions and 20k for answers.) Posts which meet some criteria are automatically deleted by a system after a certain time, but sometimes it makes sense to vote to delete manually. In fact, there already have been some related discussions on this meta: Proposal to more aggressively delete off-topic questions or Should we try to re-start manual deletions or is the situation fine anyway? Still, I think that it might be useful to leave some time so that the feedback on why the post was closed actually reaches the OP.