Thanks for all y'all's help!
I went back to the beginning and decided to just have the stars continuously scroll. The point is the player is drifting, so it doesn't matter that they don't control the direction.
You can see my changes here: https://github.com/docwhat/screams-into-the-void/tree/9153219460a57a2d4ff18804f6808f8a3824f32f
I found out my shader had some math bugs, so I went back and re-read the documentation on what FRAGCOORD, UV, etc., actually are, and that helped. I also went back to re-read about OpenGL, since I only attended SIGGRAPH stuff in college.
The real problem, though, was that I had math going out into the weeds, which would eventually cause the shader to stop working.
I kinda wish there was a way to add assertions in the shader during development. I know it would be hard since they all run in parallel and there can be hundreds of millions of them... but still.
Ciao!