The reasoning is both fallacious and just plain wrong.
Allowing anonymity does NOT make criminal behaviour impossible to detect, either the act of criminality or necessarily the individual responsible.
Even if you can't detect the individual the fact that the criminality has occurred can still be detected and addressed.
If I steal a gun, shoot someone with it and run off without being identified, I've still committed a crime. Even if the police can't identify or find me they can provide redress to my victim and assure that the person from whom I stole the gun better secures it in the future. Actions to reduce crime can be taken despite the anonymity.
So no, the reasoning is very flawed and the question of whether to remove anonymity or the classification of actions as 'crime' doesn't even need to arise.