Another thing that you can use to address this is to develop drills to force the catcher to trust their gear (which requires them to have gear they can in fact trust). We put new catchers in the "cowboy" position. They are on the floor/ground in the block position, in full gear but without a glove. Butt as close to the ground as they can get, with toes pointed out as far as they can. Have them place their hands "in their back pockets". If they are very timid, start with tennis balls. Simulate block situations where they are forced to allow the ball to hit their gear. Make them see the ball all of the way in, with their head down, and watch where it lands after it leave them (hopefully it stays close, but that isn't the purpose of this drill). As they get more comfortable, use softballs. If they are using proper gear and proper form, they will eventually realize it doesn't hurt and their confidence will boom. If the catcher isn't seeing the ball into the mitt because of being timid, this can certainly address that issue.
Yes! A couple of years ago we had a girl decide that she wanted to catch. Her first practice, we had her gear up, we set the pitching machine on a low speed and fired some balls at her chest protector and shin guards. Let her see that her equipment will protect her...