A common use of serial number is to prove ownership. If you don't have the serial number linked to your identity then that is your problem. My shop registers the serial number. If they don't put it on the receipt then force them to write it on the receipt. On a used bike write it down and take a picture with your residence in the background. My local police will let you register a bike and they will check serial number of any bikes recovered against that list. Stolen property reports are sent to pawn shops.
You find your bike at a pawn shop. If you have proof of ownership of THAT bike they will return it to you. In my state they have to hold property for 2 weeks before selling.
You suspect a neighbor stole your bike and you take a picture of your bike in his / her garage. Take that to the police and they will investigate. Without a serial number you cannot prove that is your bike.
You see your bike on CraigsList and you call the police. If police won't meet you then go to the buy take the bike and call the police.
You see your bike on eBay and buy it. Prove to PayPal you bought your stolen bike and they will take action. eBay does not want to be in the stolen property business.
A serial number may not slow them down from stealing it but it sure slows down selling it, aids in recovery, and help convict criminals.
A common scenario is they will take a van of stolen bikes to another state and sell them there. I buy a bit on CL and if they have out of town plates I walk. If they cannot give me a credible history on the bike I walk.
Another user of serial number is fraud detection. Is it even a valid serial number? Is it the correct frame for that serial number? Multiple bikes with the same serial number - definitely some counterfeiting going on.
Proof the property property was stolen. If the police raid a garage or warehouse because they suspect goods they have hard evidence that bike is stolen property.