> modern displays don't paint the image line-by-line (...) They light up each pixel simultaneously, refreshing the entire display at once.
The entire screen area is lit all the time now, yes, but refresh still typically happens line by line, top to bottom [0], left to right [0], for both LCDs and OLEDs. It's a scanning refresh, not a global refresh (sadly).
You can experimentally confirm this using a typical smartphone. Assuming a 60 Hz screen refresh, recording in slow motion will give you enough extra frames that the smartphone camera also likely operating in a scanning fashion (rolling shutter) won't impact the experiment. On the recording, you should see your screen refreshing in the aforementioned fashion.
[0] actual refresh direction depends on the display, this is for a typical desktop monitor