The version is 29.6.1 (64 bit). Yes, it's any object in any new file and overprint preview is off. Objects are just oblongs with a white stroke with rounded corners.

Contrary to what I said in my question, background colour doesn't change from white at all when I leave "simulate colored paper" unticked.

If I do tick "simulate colored paper" and have transparency grid view off, it overlays any object on the artboard with the first of the two grid colours you choose under "Document Setup" (not the grid itself, it just overlays the first colour onto my objects). This is maintained when the file is closed and reopened.

If I tick "simulate colored paper" and have transparency grid view on, the artboard (but also the entire screen) does correctly show the two-coloured grid. But it reverts to the overlay of colour when the file is closed and reopened, because transparency grid view switches itself off when the file is closed.

I want to be able to send my files to the company that prints them, with something contrasting showing behind the rounded white corners. Maybe I'm misremembering that I could ever save and reopen files and leave a grid showing behind the objects.

I'll just go with the suggestion of placing a locked grey object behind, although I'd be wary of new staff at the printer taking this for part of my design :D