This is a link to an set of jeweling tools which you need for putting in and removing rubbed in jewels. They spread the brass over the edge of the jewel to hold it in place and the other tools push the brass edge back away from the jewel to allow removal. I am having difficulty telling by looking at your picture but if you have already broken jewels trying to just push them out then it would be safe to assume they are the rubbed in type.
I understand how to use your method for removing the jewels, you would have to be careful not to remove too much brass otherwise you would have problems getting the new jewel rubbed back in. Using the hand tools I have shown above would be a much quicker method, I can remove a jewel and rub one back in in a couple of minutes.