The ball just presses into the base housing. It has some splines machined into it to bite a little into the base, but it's just pressed in. Just pull straight out with enough force and the ball will come out of the base. It's been a number of years since I did that job, and the details are fading, but I think I just grabbed the round part of the ball with some soft jaws in a vice and then tapped the base with a plastic mallet until it came free from the ball. I do remember that I used a little epoxy on the splines when I put it back together just to take up the gaps and be sure it stayed together. Splined fits like that are usually not designed to be used more than once.
So about your method... I toyed around with similar ideas with a slot in a piece of something that would keep the ball from pulling through, and gave up because I had to either a) make the slot big enough to fit over the shaft of the ball, or 2) make the reinforcement out of something thin enough to bend like you did. Problem is I was worried if I made it out of something thin, it would just bend under the force of use.
What I did NOT think of, was your great idea of using the screw posts for additional support for the reinforcing part. Using those posts to support it should keep even a thinner material from opening up again under use. Much simpler than what I did, and probably works just as well!