Earlier quoted context omitted.
If you blur then mosaic, or vice-versa, then presumably you get rid of the low and high frequencies? Depending on the detail shown in the original image either, or both, might remove enough information to render the image anonymised. How about replace each face with a "this is not a person" AI generated face, then blur+mosaic. Or just a non-person face using a deepfake system that matches the facial expression?
Why do all these complicated things? Just draw a black box over faces.
Don't tell people what not to do. Figure out why they're doing it, and provide what they actually want while still achieving the goals (here: security).
Very coarse mosaic, add noise, then blur seems reasonably safe, and doesn't have to look like crap.