If you don't tell anyone what town you're from, they can't do what you're suggesting.
That's not to say that the information can't be found, just that it makes it much harder. So we're in agreement that this whole thing is a matter of convenience.
I'm also not making any arguments directly related to this new law, I'm simply fleshing out the reason that this sort of law has come about. People in general are ignorant of how criminal justice actually works, especially on the side of convicted persons. If people did not automatically hold a person's past transgressions against them, this would all be moot; however, there is a strong assumption in Western societies of "once a criminal, forever a criminal," particularly in the U.S. and this assumption is a large driving force of the vicious cycle of criminal reoffense.
I agree that people tend to be too harsh on people with a criminal past, but I still do not feel that this law is justified (and I realize that you haven't commited to claiming it is justified either, although you seem to be somewhat sympathetic with its intentions). This is a matter of the consequences of people's perception of past transgressions translating to future transgressions. While I agree that those assumptions may often be unfair, I shouldn't lose my ability to search publicly available data and form my own opinion on what that person has done and whether or not they have changed their ways. Besides, I would hope that the ability to find people's wrongdoings more easily would act as a bit of a deterrent the next time a person considers morally-questionable behavior. Rather than making a law that prevents people from finding publicly available information, I believe we should address the problem by using statistics and success stories to educate the public about criminal reform rather than take away people's ability to learn about their past.