You can have custody information be open for query without exposing all of the circumstances, and without releasing mugshots to private sites that will extort people to have them taken down.

You can do something very simple like having a system that just lists if a person is - at that moment - in government custody. After release, there need not be an open record since the need to show if that person is currently in custody is over.

As an aside, the past few months have proven that the US government very much does not respect that reasoning. There are countless stories of people being taken and driven around for hours and questioned with no public paper trail at all.