It shouldn't be "just a search". In other words, it should be possible to go to a web site, enter an address and find out who owns the property at that address. But the same site should not allow the user to enter the name of a citizen and get the address where that citizen lives, or who else lives at the same address. So businesses who possess the data should not be allowed to just dump the entire data into Google's search indexes, they can only allow certain narrow searches.
So, again, you're basically proposing that even though data is legitimately in the public domain, somehow people should be prevented from connecting that data with a search. Good luck with that.
The government should not put citizen's personal information in the "public domain". When government collects personal information about its citizens there should be laws governing what is allowed use and what isn't.
The list of people who voted in a particular election is placed into the public domain as an anti-fraud measure. It's an important part of the US version of democracy. Other info like property ownership is public for similar reasons. Rich people form trusts to buy property and obscure their names.
And in some cases there are. Your tax records are not public. But a wide range of transactions/interactions with both the government and private companies (such as home purchases and arrests) are as a matter of law.
