Strictly speaking, that is correct. And note that the Indexing chapter in CMOS covers names from several different traditions besides Japanese that do not normally use a First Name Surname convention, including Chinese (CMOS 16.77), Hungarian (16.78), and Korean (16.82).
But it can sometimes be a challenge to determine how the name of a particular author is supposed to be styled. And as you suggest, some publishers impose First Name Surname order even for authors who clearly put the surname first (sometimes simply by adding a comma after the surname in a bibliography entry, even when the author’s name appears in the text or with the cited source as Surname First Name). And some authors who put their surname first in the context of Japanese or Chinese or the like use so-called Western order when writing in English, further complicating matters.

We’ll try to include additional information in a future edition of CMOS. But as long as personal preferences and publisher’s practices vary, there will continue to be some inconsistencies (though centralized databases for author’s names like ORCID iD may eventually help to sort things out).

In the meantime, and again, you are right that our advice in the indexing chapter is intended to apply equally to bibliographies.