Their position would increase as they increased in rank but they are still just secretaries, even at the highest level.
On a ship with a captain who was a full captain, they would be a senior NCO.
This is false. In the documentary Carrier (2008) which follows the USS Nimitz on a deployment in the gulf they follow a yeoman, Shaneka McReed for some of it. She is promoted during the episode to E-4, Yeoman Petty Officer 3rd Class. She was a yeoman who served on the bridge of the carrier. (I just recently watched this which is why I thought you were incorrect on your description).
I don’t know about rank changing since the 60s other than in 2016 they no longer referred to sailors by their ratings, only their rank.