Most of the adhesive work involved polyurethane glue. I like Franklin’s recipe. Glue in the holes, water on the spindles. There is a sweet spot in the cure for cleanup time, but I don’t have it down quite yet.
I also discovered that, even though the lower section was rigid, there were a few spindles loose. With these I upended the chair and bored a 7/64 hole into the tenon part of the rung, angled so it would hit the void. The hole is just big enough to insert the nozzle of a small tube of CA glue. I just gave it a healthy squirt hoping it would be smart enough to glue the surfaces it could find and wet enough to expand the wood to tighten the joint.
It’s working so far.
-- "...in his brain, which is as dry as the remainder biscuit after a voyage, he hath strange places cramm'd with observation, the which he vents in mangled forms." --Shakespeare, "As You Like It"