Get a book on repairing furniture. Those chairs look fairly recent and rather inexpensive. The 'carving' on the vertical back piece looks more like it was stamped than carved. The wood might be some kind of mahogany. Or maybe birch. Or some Oriental wood. I'll bet those chairs are much lighter weight than they look---an indication of foreign grown wood.

Regluing with modern wood glue is probably the easiest, but I'd bet the failure is related more to bad joint/construction fitting than glue.

And most of the old glue needs to be removed before new is applied---which will further impact the fitting of the pieces.

Epoxy might work, but I have only tried to repair a couple of old cheap chairs---without much success.

To restain means getting ALL the old finish off---something requiring a professional dipping process.

When I reglue chairs, I use hide glue if that was what was used on the original (usually pre WWII) or white PVA (Elmer's) otherwise. If it's cheap furniture and won't last to another re-gluing, I might use epoxy.

All the joints must be cleaned and tight before re-gluing.

Looks like a fun project. I would definitely strip the pieces when you have it apart. I haven't tried all the strippers but I think almost anything would work unless this is a polyurethane finish... I don't think I've ever tried stripping that but have heard it is harder to do than most. The challenge with round pieces is the physical removal of what the chemical loosens - I'd stock up on 3m stripping pads and thick rubber gloves. That and something like dental tools to get it out of the crevasses of the carving.

Karin L

First you have to know what glue was used.
If you can't discover that then, you need to remove the contaminated wood to expose fresh wood and add more wood to get the joints back to pristine. That's not that big a deal.

The reason you need to know the glue they used is simple:
Pretty much nothing adheres worth a damn to old PVA glue not even more PVA.

Epoxy, PVA, resourcinol, etc won't adhere to Plastic Resin glue and PVA won't work and play well with formaldehyde glues.

If the chairs are really old the glue will be hide - if you are very lucky. Test this my heating to 140F and seeing if the old glue turns liquid and smells bad.
Hide glue sticks to hide glue quite well.
The way to tell what the glue is is to burn it and smell the smoke. Ya gotta scrape some away avoiding any wood fibers. YA also need to have already figured what the various glues smell like when burnt.

So, you are really in quite a pickle that probably can best be solved by removing the contaminated wood and rebuilding the joints.

Thanks for all of the suggestions!

I've seen what looks like old yellow wood glue, and some remnants of a clear glue, maybe old hide glue?

Either way, I have a bottle of hide glue sitting around, and think once I clean all of the joints, I'll give the hide glue a try first. At least if it fails, I can steam it off and start over. Or pitch the chairs.