Eddie Vicarious in Northern AL
Sorry I am replying to a post that is a month old. I wonder -- did you finished repairing your Windsor chair? Joe's suggestion was a new one for me and interesting, plus the site he referenced (by a pro, no less) seems a good place to visit. All that said, I would go with the through spindles and wedge them from the top, plus wedge the back where it comes through the arm rail. (It would be a far stretch to call me a chairmaker, but I have made a few in classes with Curtis Buchanan* and Mike Dunbar. This is how we did it, and the result is very strong.) As to the splat, I guess that I would make a blind mortise for it in the back. You had already done the hard parts as of your post, I would say. Only the exciting parts remained!
Curtis Buchanan has utterly fantastic videos on YouTube showing every step in making various Windsor chairs, and Peter Galbert has great online resources as well -- if you were to need help moving it forward. Plus, you could ask back, obviously.
Really neat chair, by the way! I'm jealous. Odds of finding such a thing in my neck of the woods: one in a million.
*If you're interested in Windsor chairmaking, put this (class with Curtis Buchanan) on your bucket list. And don't wait too long! It was easily the best week of my woodworking life -- by far.
Show us how your chair turns/turned out, please.