The touchscreen wasn't responding well. I bought a new display assembly to replace the old one.
Turned out the problem is probably with the digitizer connector on the logic board. There are a couple of little metal connections that seem to have broken, and the touch screen is consequently unresponsive in certain places. iPhone bricked. Oh well. I'll save the replacement display for when my wife's SE breaks.
If your connector/port on your logic board is broken, a new screen won't help much.