I worry that contributors who aren't familiar with the idiom would copy method signatures that don't have the const-ref, but not know to copy the move in the implementation. Preventing that is a bit of extra work for reviewers.
The back-and-forth in your bug nicely avoids that, BTW - the reviewer reviewer challenged why you didn't have a const-ref and you were able to explain. It would have been a red flag if you'd responded, "I just did it that way because SetFooString did that. Isn't that the normal style?" If reviewers are no longer specifically looking for "const string&" they would never have asked, and might have not noticed if the implementation was wrong.
So I prefer to keep "const string&" as the default, because it's easier to review since all the information is in the signature, not split between the signature and implementation. And because of that I would avoid std::move unless an individual use is shown to be a win for performance, even if it is safe as a default, because having two ways to do the same thing is extra load on reviewers.