If you look at the now exposed joint you can see how little wood remained after the original dowel joint was constructed.
This will give you an indication of how little tolerance you have for installing new slightly larger dowels.
I THINK that part of the problem with this chair was a split in the leg -- you can see where dirt / finish has penetrated the surface opposite the back rail.
To repair
clean up the exposed dowels by carefully scraping the residual glue from the exposed surfaces. Including the other half of the dowel holes.
use painters tape to mask off the finished surfaces.
apply a coating of PVA glue to both split faces.
Put the piece back in place and secure with painters tape tightly wound around the joint.
Mark the repaired chair as "for looks" and try to avoid using it. And when you do use it place it for a light weight person who won't rack back on it.
for future chairs, if the joint won't tap apart easily, cut the dowels.