On the off chance you are not really a troll:
"Surely their whole purpose is the protection of vulnerable individuals"
No. If that was the case then it would be OK to assassinate any non-vulnerable persons "just to be on the safe side".
Their main purpose is to protect the vulnerable, but with a duty to not screw up anyone's life who doesn't deserve it.
"A false positive means they did the paperwork wrong, and likewise a false negative means they let someone through who records showed was a risk."
And yet you fail to see the realities of this. Let me re-write it for you:
"A false positive means they unjustly tarred someone with being 'unsuitable' for a sensitive position and may as well just tattooed "paedo" on their forehead, and likewise a false negative means they put some vulnerable persons at an increased risk through failing to do the very thing they are there to do"
Importantly if there is [believed to be] a valid reason for a CRB then if you are seen to fail it then mud will stick very badly, and if you have a situation where there is a risk to one or more vulnerable people then there is a duty of care to make sure the bad guys don't pass through.
"Do we know that a false positive was really no risk? Of course, we don't."
Um, not the sharpest tool in the box, are you? Couple butties short of a picnic? Tinny short of a six-pack? The crb is not able to identify a risk through clairvoyance, it is supposed to check someone's background and make sure they have no previous, whether real or insinuated.
To use a "every man is a potential rapist" type argument is frankly absurd, your reasoning appears to be "if they fail the crb then they are a risk, and if they pass the crb they they are prolly a risk anyway - they have just not been caught yet". You'd never employ anyone