I don't know if this is the specific problem that the person you are responding to was alluding to, but: a purely anonymous system is going to end up with hiring tons of young white males--surely fewer than a company that actively discriminates against other groups, but noticibly more than a company that is actively looking for diversity--due to young white makes making up a large percentage of the available talent pool at any given level of expertise (due to a combination of factors, including but not limited to all of "available time", "increasing population", "availability bias", and "systemic oppression"). A company might specifically find value in "we don't just want the world's best engineers: we want a diverse set of backgrounds and viewpoints", and a forced anonymous hiring process actually removes the ability to do that, which might actually help continue to entrench the systemic oppression issue (as then everyone builds things for the same market demographic as that's all they understand).

Anonymity would not affect diversity, but it would not necessarily enforce it. Indeed until all races, genders, sexual orientation, socioeconomic strata, etc. apply in identical numbers, one must discriminate in order to guarantee diversity.

EDIT: Upon further reflection, anonymity could help combat ageism (scrubbing specific graduation and employment dates, for example). but in the general sense I believe the point still stands.

How do you anonymously interview for cultural fit?