I do a lot of lith printing on when-they're gone-they're-gone papers, and I'm also used to people saying "wait, they still make FILM?" "You do that red-light room stuff, like in the movies??"
So I like to treat some prints somewhat like "this is an artifact". Like, mount a 16x20 Ektalure print to cotton rag paper just a hair narrower than the print (to keep it stiff but still a sense that it's a sheet of paper), and then float it a hair in the frame (like foam board behind, and hidden by, the print, so it 'floats" in the frame). There's the 2D image, and then there's the 3D "print" itself - 2 levels of visually experiencing it. An image and an artifact, so to speak.
When matting, I really try to leave like 1/8" of printing paper visible, for the same reason - "this is a print on paper", not just an image. Some prints, I may decide I'd like them cropped tighter and so I'll lose the print border.
I used to print everything with a larger bottom border, sign and date the print itself, and matte with 1/8 to 1/4 all around, with extra opening at the bottom, treating the signature as part of the print. Some prints that looks great, some it looks "off" - never really figured out what makes it work on one or not the other. So lately, I leave a thin border all around, and sign the matte.
I do clean up my borders regardless of matting - any testing stains or (lith) fingerprints and generall what-the-hell lith border stains - bleach them off. Kind of figure if someone ever re-matted the print, they won't see a total mess under there.