Meaning if you could prove police arrested someone and they could not produce a mugshot, something is wrong. Providing them a way to deny the mugshot without being suspicious gives them more options.

Couldn't the police just take a mugshot, confirm the arrest, and say "this person was later released without charge, none of our business where they are now"?

This happens, but between most people having friends/family and high profile cases having groups outside the station, this causes problems for the police. The proposed change gives them more time without needing to deal with those problems.