Thanks lazygardens. Yep, when I opened it up, I couldn't believe how this thing was built - it was NOT inexpensive, but clearly a Chinese product stamped out as cheaply as possible. Lipstick on the bulldog if I ever saw it. So, for clarification on your suggestion; do you think that I should leave the base "as is" and just glue/screw straight into the legs and round frame? I'm worried that if I don't do some work to stabilize the legs first, they will just split out or become loose again, as a single screw would be bearing a lot of shearing force as the chair is used (leaned back in by kids, etc). Let me know what you think.

I have used Gorilla Glue successfully, & it will work OK for the repair you propose. Be aware that this type of glue expands into a foam as it cures, which requires that the two pieces of wood be firmly clamped. If there is no clamping, the glue will foam up and actually push the two halves apart, leaving a very weak, and ugly, glue joint. You might be able to wrap the armrest with many turns of heavy twine, to clamp the assembly while the glue cures. Epoxy glue is very strong, and does not require high clamping force. I would very likely use "one-hour" epoxy on an armrest like yours. The damaged crossbar could be sanded out and varnished. I'm not sure that staining will be necessary. I doubt that the repaired armrest will look original, either. It will need to be sanded to remove glue residuals, and then it will have to be varnished. It should look pretty good when finished, but not exactly matching the original color.

Wood joints need some clamping pressure to work well when glued. You can use clamps, tightly tied rope, or even weight. If the rockers are left out year round and were designed/built to be indoors furniture, the temp/humidity changes may be too much. Try the Titebond as recommended with clamping pressure---cures in 24 hours. If the joints still separate, get different rockers.

Here's a detailed video on how to repair loose chairs. We do this as part of our business and we freely share our knowledge so you can do it yourself.