"The screen seems fine; so does the backlight."  Though I doubt the answer is "yes," does this mean the PC is functioning normally, except to give lines and grids at startup?

AIO's in general can be more prone to heat issues due to the tiny fan(s) used to cool them.  The monitor may have been affected by this.

Test w external monitor via the rear DP video port of optiplex AIO.  If you get normal video on external monitor, then mobo video adapter is good but screen or video cable is bad.
I have seen Dell laptop screens go rogue.  Chances are the lcd screen is bad but not display cable,

@bradthetechnut​ can you provide the screen part number.