where's my table saw?
The part that broke is called a spindle, or stretcher. The end may be called a tenon.The right way to fix that is remove the tenon. Use a heat gun to loosen up the glue. Careful not to hold it too close or too long or it will damage the finish. If it’s over 75 years old good chance it’s hide glue.
If the tenon is stuck that’s a good indicator it’s well built.
Orient the grain in the stretcher vertically for maximum strength against a foot..
This is the right way to do it. Use a heat gun to soften the hide glue. Wrap the short piece with several layers of blue tape and use a channel lock pliers to rotate it slightly when the glue is soft. Then it will pull right out, some wiggling required!
That break is clean enough to glue back together. The issue will be how to clamp it because it's round.
Take a 1 /12" square block 3" long or so, and drill a 1/2" hole down the center. Then saw it down the length and through the hole. Now you have a flat surface to clamp against. OR you can wrap it with electrical tape, because it will stretch as you tighten it around the break.
Sprinkle a small amount of salt in with the glue to keep the joint from sliding around when you apply clamping pressure.