Here's what we should do: disband NASA and set up a land office to register space claims under US sovereignty.
One caveat. To keep people from selling the "Brooklyn Asteroid", or claiming to be King of the Moon, you have to be able to stake a claim physically on said resource.
Think that'd work?
There is already a huge legal base in case law to handle shysters and scam artists--ursurpers--once property rights are established.
The point of private property rights is establishment of collateral so investors will have reason to invest. Right now, they don't have reason to invest except at the Allen hobby level. Serious money awaits private property rights.