The US Constitution is not common law. Common Law is only binding on the Federal government to the extent that it supplies the definitions of words, phrases, and legal concepts that are actually explicitly in the Constitution. You "can't say "Common Law principle X restricts ordinary people, therefore it restricts the government." That just isn't the case.
The airplane analogy is quite instructive. Just as I am not legally allowed to fly a plane, the government is not legally allowed to create certain massive databases*. But since other people are allowed to do these things (ie: fly a plane or create a massive, intrusive database), both me and the government can pay them to do it.
As I've said before, this is actually the biggest security hole we've found in 200 years of Constitutional Law. It's trivial for a bad actor in government to get someone outside of government, and thus not restricted by any of the Constitutional bans on governmental oppression, to do the dirty work. He won't be as efficient as the Gestapo or KGB, and he needs some help from all three branches; but that doesn't mean he can't actually do his dirty work.
And the problem is in the basic architecture of the document. It can't be fixed without an Amendment or a bunch of Judges who are convinced the country will be much better off if they BS this one.
*The Census Bureau and the IRS are examples of perfectly legal extremely massive databases.