Hello Kyle,
I tried the above, but it did not work. I had set up a "control field" i.e. used the same field on a visible tab in profile position.
The "regular" output did what it was expected to do, the field disappeared.
The substitution however did not disappear if I put it inside a "fields delimiter field".
I did try it with a second user, did not work, so it can be ruled out that it´s an issue of user access/me not being logged out and seeing the items set to "private".
On a side note, I have noticed that the privacy options don´t appear or work for "fields delimiter" fields on user profile edit. Is that on purpose?