Without being able to see it (and I'm no expert anyway), it's hard to recommend anything specific, but whatever you end up using (glue, tape, etc.), make sure it's labeled "archival". That'll ensure it won't discolor or whatever down the line.

I bind my own books using fiberboard and leather, and wood glue has always worked for me. My oldest book is around 9 years old and it is still going strong

I have a related question. I have some very, very, very nice editions of books that are in pristine condition. They were held in protected cardboard boxes and the boxes did their job well while the books were in long-term storage for almost a decade. But the boxes themselves are in very rough shape. The actual surfaces are fine (except for a minor scuff mark on one, but I already know how I'm going to get rid of that). The problem is that the lids are coming apart at the corners, turning the lid into a flat piece of carboard with four flaps instead of, you know, a lid.

What would be the best way to repair those corners so that it looks at least passable to casual inspection. The boxes are cardboard covered with textured black ... something paper, but not card stock, nor regular paper. Where they're torn at the corners, the card stock, no longer contained by the black covering layer, has kind of, over the years, puffed out and gone feathery, so even if I glue the corners back together with something, they won't be that nice textured black all the way.

Does anybody have any ideas how to repair this, or should I just embrace the look of covers which did the perfect job of protection and look like wounded warriors or something?

I'd recommend going to a nearby library, asking the librarians about their own book mending process and getting their advice. Most do their own book repair.

a hard cardboard sleeve

So is it more like a slipcase, then?