There's not much lateral play, more rocking back and forth play when it's partially threaded. But that goes away when I snug it up.
I figured out a little more. I couldn't tell what exactly it was snugging up against while threading. It turns out that the snug position is the position I unscrewed the chuck to before finding the set screw. So the messed up spindle threads are preventing the chuck from screwing in all the way.
This should mean that if I clean up the burrs on the threads, it should thread on all the way. And hopefully there are enough good threads on both the spindle and the chuck to give a secure fit.
I could clean up the threads with a file. A die would be better, and $16 would be a bargain to save the lathe. And from the reviews, it sounds like this is the main use of this size die!