This makes me want to ask about my Windsor chair. It’s back wood is thin (3/4” & 3/8”) and in two places has broken. Repair or leave alone. I usually put something in the seat so no one sits in it. Family heirloom (we didn’t break it).
It would depend on the break. Modern wood glues are incredibly strong if applied properly. Send me a PM with a pic when you can and I’ll have a look.