"So in your view, upon being mistakenly arrested - because police made a mistake - you have to go through the hassle to being removed from that database?"
Yes. I know that seems wrong, but surely you want the PNC to record the activities of the police as well as criminals? So they should have to record any arrest on the system, regardless of outcome. Now consider data integrity - editing rights should be severely limited, to prevent data being improperly edited by the incompetent or the malicious. So the officer or clerk who enter details of my wrongful arrest should not be able to delete that, merely add to the data (no idea how PNC works in reality). That means we'd need a data controller for the PNC, with an appeal and deletion process. If the local nick are busy arresting you without due cause, can you trust them with record editing rights? And if incorrect arrest records were routinely deleted at the local nick without any process, where's the downside for police officers from making pre-emptive arrests because it suits them?
Even then, if you were stopped repeatedly by the police, and believed eventually that this was intentional harassment or discrimination, if you've actually had the details expunged, where's your evidence when you start to see this as something more systematic? How would you hold police forces to account if they had a high rate of wrongful arrests, or released too many people without charge who might have been chargeable? What about events like domestic violence, where it is common for the victim to call the police, for the offender to be arrested, but the victim then refuses to press charges. That can build, the situation can worsen, and knowing that there is a history is very useful in trying to react to future instances; It also helps when there's a case conference, for example, that looks at the interests of children at risk of domestic violence. In fact, more widely, it becomes a problem that if you can frighten witnesses enough to avoid giving evidence, not only do you avoid a crmininal record, but you force the police to delete all records. Not very sensible is it?
There is no good outcome here, I'm afraid, but the idea of simply deleting records where no charges are brought seems to have its own downsides.