Best advice not followed quite soon enough: "Dust collection is the FIRST consideration when setting up your shop".
There's so much poor advice on the internet it is hard to choose but, a compound miter saw proved to be the most useless investment this side of a biscuit joiner I have ever been told that I "need".
Oh, and my favorite: "I been using a table saw fer forty years and ain't never needed a guard".
Old-timers and experienced hands that don't over-emphasize safety to newbies are just plain irresponsible IMHO.
I'm going to guess that this may be a reaction to the fact that you have a structural issue. They're both structural but the one that makes me cringe is here:
For the long crack I would be tempted to clean it out carefully with a folded strip of 400 grit sandpaper taking care not to further damage the top (show) edge and blow it out with compressed air. Run some painter's tape on the seat surface along the edges of the opening where you are going to get squeeze out.
Spread some decent PVA glue (yellow woodworker's glue) in the crack (you can squirt it in, draw it in with a strip of paper or piece of thread in the tight spots), and clamp it over night. That's more or less how the joint was done in the first place. Abuse caused the failure, not the assembly method.
For the crack at the corner where the upright penetrates we have a more serious problem. Patience is your friend on this one. The potential leverage on that point is fairly high. . . . I'll be back . . .
Darn, I don't have SketchUp on this PC so I'll have to try a description. I am not opposed to a mechanical assist in these situations. You could clamp the crack closed, drill a hole perpendicular to the edge, between the back edge and the exposed through-end of the upright. Sorry about the make-shift diagram.
The long hole should be sized for your pan head or round head screw (not flat head). You will also want a counter bore as you will use a plug to hide the screw head. If you use a #8 screw with a consistent diameter like so . . .
. . . the counterbore would generally be a 3/8" hole about 1/2" deep. You don't have a lot of material to take advantage of so your counterbore should be as shallow as possible; just deep enough to let the head recess with enough counterbore left to hold a plug.
Using the dimple at the bottom of this counterbore hole, drill an 1/8" hole the full length of your screw shaft (I'm guessing 3" by the pics).
Remove the clamp so that the crack reopens. Put glue or epoxy in the crack and re-clamp. Run your screw in recessing the head below the chair's edge surface. Clean out any squeeze out in the counterbore, you'll want room for the plug. Leave overnight.
After coffee the next morning, remove the clamp, clean up any glue issues and pick up your piece of 3/8" dowel or hardwood plug you acquired while waiting for the glue to dry. Run some glue in the counterbore with a stick or Q-tip, tap the plug or dowel in with a hammer and let the glue set.
Saw or sand the plug flush. It should be fairly unnoticeable when you finish is applied except to you. You will know where it is and think it is a glaring red light with arrows pointing at it. Don't let that bother you. We all feel that way about the minor imperfections in our work ;-)
That looks like glue line separation. That chair may be falling apart and not "splitting." The Dutchman concept may not be best, but all you lose is time. I would try to disassemble the chair enough to clean up that joint and re-glue, but chairs are very tricky. Just thoughts. Don't want to scare you.