How do I fix interlacing lines going up and down on iPhone SE display
As it turns out, it was something in the display assembly. I purchased an iPhone SE LCD Screen and Digitizer Full Assembly from iFixIt, then followed the meticulously-detailed iPhone SE Display Assembly Replacement guide, and after another hard reset, the display was back to its former glory!

One caveat for those replacing the display assembly: When re-assembling, if it's hard to snap the top part of the display into the body of the iPhone, check if there are extra bits of glue or plastic on the new assembly. It seems the iFixIt replacement parts sometimes have an extra bit of goop that needs to be shaved off in order to fit well.

If the display doesn't snap in snugly like the old one, you could end up putting pressure on both the earpiece and vibration motor, which can make sound and vibration features a bit strange (tinny or even nonexistent!) until you fix the problem.