"And with the power of today’s computers, it would be easy to keep track of weapons by using serial numbers. Caught with a gun, the serial number of which has been removed…go to jail."

It would also be very easy to create a centralized registry under that scheme, which could be used at some point in the future to confiscate guns from lawful owners. Guns can't be confiscated unless the government knows who owns them, which is why the anti-gun people are trying their best right now to establish a back-door registry. BTW, it is already illegal to alter or remove a gun's serial number and you can go to jail for that. Existing law also already covers improper transfers and negligence.