If you can find some methylene chloride paint stripper, that will be easy work. The MC is dangerous and the EPA has outlawed it, but the population has somehow survived a few generations when it was available. The newer stuff works, just takes longer. I partially agree with tipsy in that you will need stiff brushes and steel wool at some point. 0000 wool will load up with one swipe so get some #2, some #0 and maybe some 000. For me 0000 is for waxing and polishing.
That chair is what is known as pressed wood relief. It isn't carved, the design was pressed in with a mold, heat, steam and a whole lot of pressure. It's a mass producing technique. And yes, you may very well need to glue some spindles if not more after stripping. Try not to use any water which will dissolve old hide glue.
There are plenty of tutorials on YT that go over regluing in place or taking apart. Some involve injecting CA glue with a crafts needle. I do a lot of that. Drill a 1/16th" hole right in the joint and fill it up with glue letting gravity take the glue to the bottom and all around the joint.
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