So it turns out that NAA and VLQ flags have a 15-minute period during which the community is given a chance to handle them before mods do. When this happens, the flag does not show up in the flag queue but it is visible in the post timeline.
The confusing part: The flag-details bar still shows up at the bottom of the page, but contains no content — just a close button. In the screenshot below I've increased its height significantly to make it clear:
Could this be removed, or fixed to indicate what's happening so that no one else flips out like I just did? (See a prior revision.)