Repairing a Tonk revolving piano stool

First off, why do people think I can repair furniture just because I make furniture? Anyway, a friend has presented me with a Tonk piano stool. One of the rungs has splintered off, leaving a splintered tenon in the center column mortise. She says this has been previously glued which has failed. At first glance, it looks like I will need to implant a dowel into the rung and remove the splintered tenon. I have tried to (gently) remove the rung and side assembly from the bottom piece but it won't come out. I guess I could just clean out the old glue, re-glue, clamp and be done with it but I don't think this would make a proper repair. I wouldn't touch it as it may be an antique, but it has had previous repairs done on it and probably worth under $200. Any guidance?