Why do the ships fly so slowly?

I assume that they're traveling VERY fast, but it's hard to visualize their speed as there's nothing close enough to the ships to provide a visual reference.

I don't know if Kubrick wanted the ships to look slow because he wanted the viewer to feel that the astronaut characters were trapped in a claustrophobic environment, or if he just couldn't figure out a good way to show the real speed of the ships. Probably the former.

"Why do the ships fly so slowly?"

What makes you think they're flying slowly?

"I assume that they're traveling VERY fast, but it's hard to visualize their speed as there's nothing close enough to the ships to provide a visual reference."

Exactly.

"But there is something close enough to provide a visual reference. The edges of the frame of the camera. Which shows that the trip is travelling very slowly in comparison to the presumably stationary camera."

The camera is not a reference. The stars or Jupiter would be a reference but are so far away that it gives the impression of slow movement and this was intentional on the part of Kubrick. This is not Star wars :)

Imagine that you’re in your car, tooling down the freeway at 70mph. Another car is in the next lane doing 75mph. You're both traveling pretty fast, but the other car is only passing you at 5mph.

Why do you say they are "flying" so slow anyway? "Flying" in space is disorienting since there is nothing closely to reference your speed with. Remember they were in hibernation for months and months as well.

The Earth seems to be moving slowly as well, but we are really moving about 19 miles per second, and we do not see any planets moving visibly.

And how is any of this a flaw in the narrative, or, as you put it, a plot hole? What is it with this obsession over plot holes? It’s like, “Gotcha! I spy a plot hole! I win!” It’s a VERY annoying trend.

Consider this is visual fictional storytelling and not a documentary, that some artistic choices are meant to evoke feeling, set a mood, or just look pretty, not transmit rational information. To be clear I'm saying the ship is moving slowly because it looks majestic and it's not a plot hole.

Why do the ships fly so slowly? - They do not.
Is this a plot hole? -- It is not.
Or is there a reason for them to be moving so glacially? --- OK, lets give an example of why this is meaningless.
Ever thought about the International Space Station... Just sorta hanging up there in orbit, apparently motionless? Cuz that's how it looks in footage of the station from anywhere in it's vicinity. But you know that is not what's happening. It's zipping along at 17000 mph or some preposterous mach number if you like those... say about mach 30.
It is always about frame of reference. In interplanetary space you are well away from anything. The stars just maintain their positions, moving only if you rotate or tumble... The sun and any planets are only going to move at a crawl that you cannot perceive - again, unless you rotate or tumble. if you want to visualize a spaceship from a point of view where it seems to zip by, you just establish your viewpoint at a different velocity. You want the ship to go by as fast as a fighter jet; just view it from a point traveling 1000 mph less than it's velocity(or 1000 mph more than if you want it to zip by in the other direction).
Ultimately the point of showing the ships "flying slowly" is that they are in a sense flying slowly... It takes many long boring airline passenger hours to get to the moon, and you can hardly even see it get bigger most of the way. Flying to Jupiter takes months and months... The trip is inevitably tedious. Should the Discovery look like it's zooming?