Chairs are well known for being tricky to mend. The only way I know to get a successful "proper" mend is to clean the joints back to the wood, and then "remake" the joints to fit as originally intended. This can include shimming the joints out with veneer or other thin pieces of wood, filling a worn mortice with suitably oversized wood to give room for a mortice, and then recutting the mortice in it, or replacing the tenon with a suitable piece of wood let back into the member that carries the tenon. Sometimes the parts need remaking, which then leaves the problem of refinishing.
Sounds like a lot of trouble? It is. That's why "ordinary" but perfectly otherwise perfectly good chairs end up getting scrapped, or with bodged (and useless) wood screw repairs. If you don't mend them properly, they just come apart again.
Incidentally if the chairs are antiques and have value, any repairs should be done with scotch glue, as they were originally.
Otherwise I just use normal PVA wood glue.
