According to Dan Hartung :
Now what has happened:

1) the seat actaully split where one leg was embedded. A sliver about
3/4" wide and 1-1/4" long is bent out of the wood. I imagine it can be
glued and tapped back in place, but will it have strength to hold the
leg now?

2) one of the spindles split at the base, not a clean break either. This
is probably ideally fixed with a dowel, but is a "for now" fix with glue
acceptable if the rest of the back is properly glued and put back together?

You can do virtually anything with the proper glue as long as the wood
itself still reasonably solid, and you spend enough time piecing things
back together. Modern glues (especially good carpenters glues or
regular epoxies, not "5 minute" per-se) are stronger than the wood itself
when used right.

Whether the result is "acceptable" is going to depend on a lot of things.
Like, how much is the chair really worth to you? Will you manage to avoid
abuse beyond its capability? How well built was it in the first place?

It's rarely something that anyone else can judge for you sight-unseen.

We've had some chairs we've bothered repairing only because the teddy bears
who use them have promised to treat them _very_ gently. One bear refused,
and he has to sit on the floor instead.

If it's otherwise reasonably sound and has some redeeming characteristics,
we usually repair it. But, turning new legs, for example, may be a bit
beyond what you're interested in doing.

A badly-split leg on a chair is probably beyond saving, unless you're going
to restrict it to light duty, or spend the time to use dowels or
make a new leg.

[He says, with a chair in the garage waiting for him to turn a new leg.]
--
Chris Lewis, Una confibula non set est
It's not just anyone who gets a Starship Cruiser class named after them.