There are a couple of cases where the automatically-generated "middle initial" on private people is unhelpful or incorrect. One, which has been previously asked about, is the case where the person uses the middle name as preferred. Another is when the preferred name is initials, which is my case. It's frustrating to see my profile give my full name as "JD D.", and this seems like the kind of edge case that an option to suppress would be useful for. Also, this ambiguity can cause confusion: for example, I was welcomed to WikiTree under the name "D.", twice.
The "trusted list" message says that someone on it can view my "full middle name, D.", which is technically true but also applies to my first name. In the case where the middle name is preferred, this message is completely wrong, as it tells people they can view the full middle name that they actually already know (and does not indicate that it is the first name which is hidden).
I recognize that names are complicated, the current setup is a compromise, and that too many options can be an interface problem, but this seems like a simple enough fix and a historically common enough situation that it would justify adding an option or two. We already have a toggle for "no middle name"; I think that could be expanded into a hidden "middle name options" menu, or failing that, two new options for "prefers initials" and "prefers middle name" could be added, since both of those configurations encourage people to enter information incorrectly in order to achieve the correct display result (e.g. I could put my first and middle both in the first name field, but that would be wrong from a data perspective). Two options would be better than a "drop middle initial" option, which would not be intuitive from its name and therefore prone to accidental misuse.