If you look at the series of images DART took as it approached Dimorphos, the asteroid jumps around a bit as you approach. Does this mean small corrections were being made right up till near the end? Assuming that's correct, why is my intuition wrong that you should be able to get it perfectly lined up from further away than that, considering the predictability of the space environment?
This is an extended comment to your self-answer, so I'm making it community.
From your answer,
As to why they weren't able to do a single, early course correction and just glide smoothly in for the last hour, I think perhaps I just expected too much.
I think perhaps you not only expected too much, but that you expected far too much. I would not be at all surprised if the ephemeris for the asteroid was off by kilometers. DART did not go into orbit about the asteroid. It instead went straight for the kill.
From your question,
Assuming that's correct, why is my intuition wrong that you should be able to get it perfectly lined up from further away than that, considering the predictability of the space environment?
Your intuition is wrong because you must have assumed incredible accuracy for where the spacecraft thought it was and for where the spacecraft thought the asteroid was. I would not be at all surprised if the ephemeris for the asteroid was off by kilometers. I would be very surprised if the ephemeris was off by hundreds of meters or less. The spacecraft hit a target about 1/6 of a kilometer across, and from the video, it hit it dead center.
Finally, regarding why you aren't able to find much detail regarding how the DART guidance, navigation, and control works, the answer is on the Applied Physics Lab's description of DART,
In many ways, it was like developing a self-guided missile — a process APL has a rich history of doing.
This is perhaps the quintessential example of dual use technology. The technologies used by DART are indeed very similar to the technologies used in a self-guided missile. If not classified, the technologies used by DART are definitely subject to International Traffic in Arms Regulations. The descriptions of DART are intentionally dumbed-down.
I'm revising my answer based on the comments and on the community answer started by David Herman.
Corrections were not made right up until the end, but stopped with 500 miles to go. This is according to the Applied Physics Lab's description, cited in the community answer. According to this NASA site, the last frame where you could see all of Didymos was from a distance of about 570 miles. So, by the time Didymos was no longer visible, corrective thrusts (and this includes changes to orientation as well as translation) should have stopped.
I think the most plausible explanation is the one given by a Reddit commenter that it was inconsistent auto-cropping. The link I gave isn't an authoritative source, but the whole reason they stopped thrusting was to provide a stable imaging platform, so it seems likely they would have gone to some effort to prevent some kind of camera oscillation from continuing.
I thought for a while that "wobbling" would be impossible because the craft is subject to conservation of angular momentum. But the solar panel arrays are quite long, so the camera could wobble relative to them. But the first article I mentioned says that they worked hard to minimize that. So I think it's just cropping.
The Reddit link has a nice video, by the way. And the first article mentioned is worth reading. It has a simulated video that actually has corrections continuing until well after the simulated Didymos is out of the picture. So a possible alternative hypothesis is that thrusting was actually occurring for longer than my previous analysis suggests. I don't have a theory on how that could be, except for my general lack of understanding.
As to why they couldn't do one clean, early burn, I think the community answer covers that well. Fundamentally they never knew the relative positions well enough to make that calculation.