I don't think people will accept this. For people with other means of payment, the main function of cash is anonymous transactions. There is no other way to anonymously transfer money. The article incorrectly lists bitcoin as being anonymous when it's quite the opposite (though with some finagling can approach the functionality of anonymity.)
Also, in your example above, who's to say that Alice didn't give the money to Joe who then gave it to Bob? Or that Alice didn't buy something legal from Bob? Simply having serial numbers and tracking the money is not enough if the money can be transferred without the transactions being recorded (like they are at a bank). That immediately makes it a lot less useful for law enforcement.
Also, in your example above, who's to say that Alice didn't give the money to Joe who then gave it to Bob? Or that Alice didn't buy something legal from Bob? Simply having serial numbers and tracking the money is not enough if the money can be transferred without the transactions being recorded (like they are at a bank). That immediately makes it a lot less useful for law enforcement.