You will need to find out how old the piece is. My guess is that this is about 1900, and that it was originally glued together with hide glue. I can't tell if the piece has been refinished or not, but I suspect that it has. It could be that the stripping and scraping of old finish has loosened several parts. Also, I have seen some pieces of furniture held together with hide glue, that has slowly dried out, and turned to powder, over the years. This takes about a century. Please don't try to make repairs with nails or staples. It should be possible to make a successful repair with glue. You will have to fill any nail holes, and if some of the wood has been cracked or split, it will be necessary to glue the wood back together.

It's been appraised: as I recall, the appraiser estimated that it had been built in the '20s. I don't believe that it's been refinished, but I'm no expert.

Sounds like glue is the way to go: I will use it and not staples/nails.

Urethane glue foams and expands as it sets up, and therefore it would not be the correct choice for this type of repair. You have to clamp the wood very securely when using urethane glue, in order to keep the glue line thin. Since hide glue was very likely the original adhesive used, it would be the type of glue that I would use for this repair.

That piece has not been refinished.
Remove the residu of the old glue by carefully scraping it maybe softening it with vinegar and finally sanding.
Replace using furniture glue....Elmer's will do fine. And clamp using either a pipe clamp or a strap clamp and padding the piece well.
Remember those pieces are not functional and will not be subject to any stress.
It's very lovely!
Linda C