This is actually much more complex that it seems at first glance. Why are arrests public because nothing is more dangerous than secret arrests, disappearing people, so they have to be public, very public, to protect the individual being arrested.
I think paying a small fee for a record check at a local courthouse or being able to call a precinct and ask was 'so and so' arrested is public enough. We don't need Slammer magazines at the gas station shaming people for tabloid entertainment or the local news site pushing them all online so thousands of locals can browse looking for coworkers. The arrest itself, being made excessively public, is enough to ruin someone regardless if it is BS or not. I've gotten arrested for absoultely stupid shit before and the cop just threw whatever he could at me to see what would stick (which subsequently got thrown out at court). Thankfully that was before this crap went online.
After a number of decimal places, nobody gives a damn.