After much digging, I think I have nailed this down.
My launcher, Before Launcher, has a notification filtering feature. I didn’t realize that notifications would have anything to do with this, but as soon as I allowed Symfonium through (no notification filtering), downloads are going smoothly without any intervention.
Thank you for all the replies and suggestions, and thank you for making Symfonium!
Edit: Also, wanted to mention that finding the “Stay awake” developer option to keep the screen on forever while charging will make the rest of this process totally hands-off.
Edit again: The “Stay awake” option is not even needed, downloading is proceeding smoothly without the screen on.