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I lived in GZ for five years, never once bothering with the police registering BS.
Without that slip of paper you are opening yourself up to a potential world of extortion. I think you’re absolutely right that no one at the PSB really “cares” about this formality. But when you leave the country or renew your visa, you can bet that you will encounter someone who will take the opportunity to collect free cash, because they are entitled by China law to do so.
There are many stories of foreigners having to pay thousands of RMB for not registering at the PSB like this: Talking Your Way Out of a Visa Fine.