That isn't the way furniture is normally made. What you have is some of the worst made furniture I've ever seen. It's unimaginable a factory would turn out work like that.
I can't imagine a shop re-mortising those parts for you. Not only would it be more trouble than they could charge you for it they would feel liable for the joint. You could take a hand held electric drill with a brad point drill bit and go deeper with the mortise and then chisel it out square. Then make a homemade domino to fit the mortise. You just have to be careful to keep the hole with square sides.
😮 Wow, I had no idea it was so bad. If the chair hadn’t broken and I hasn’t found this forum, I would have had no idea it was so badly made. I questioned what I saw (shallow depth of joints and how they weren't squarely placed in the middle of the wood pieces) when I was drilling out the broken pieces but I was totally taken in and fooled by “solid wood constitution,” “no assembly required” and of course, the cost of the chair <sigh>. How is someone supposed to know if the furniture is properly made or not if you can’t see the joinings? I’ll go with the drill bit and chisel for now and look into the Beadlock system that Catpower told me about. Thanks again for helping me with this.