Why does the Moon appear to be flat? There's no way around it: when I look at the full moon at night it looks like a mostly flat disc, with at most a bit convexity in the middle. Phases of moon look like a full moon seen through a cut-out.

Why is that the case? I recall, when I was really young, looking at it through what at the time seemed to be a very strong telescope, and it appeared not only a lot more textured but also a lot rounder. A similar thing can be seen in a video I saw a couple months ago, at 2:05: https://www.youtube.com/watch?v=XCrJ3NflOpE . So it's clearly a naked eye thing.

I presume it's related to the optics, which made me hesitant to post it on this stack exchange (thinking it might be more suited to the physics one), but chose against it presuming people here would have more experience with celestial bodies and "large" distances.

I have a hunch that the answer is very simple, but I really don't know what that might be, especially since it's done over what I view as large distances.
It is an optical illusion.

We perceive nearby objects in 3d because we have two eyes. As we see objects from two different viewpoints, our brain can put the images together to make a 3d image.

Objects that are more distant perceived as 3d if they are moving fast enough for us to see them change appearance. Again it is not in the eyes but in the brain that the perception of depth occurs

The moon is a long way away, far too far for any 3d from our two eyes, and doesn't change fast enough for to see any motion. It is too small for us to see shadows or any other clues that we could interpret as 3d structure, so it looks like a small  flat shape.

Telescopes can make shadows visible, and time-lapse can make the motion of the moon apparent, and so add to the perception of a solid.