The stuff that people put in space should be seen as assets with corresponding liabilities. So, I see nothing wrong with involving the private sector as the private sector often does a better job of maximizing the return from assets while reducing associated risks and liabilities.
NOTE: The private sector fails when that system allows private companies to externalize their costs and shunt off their liabilities onto the public.
Liability in space is a very particular case, Space is under international law and the Outer Space Treaty, signed in 1967 and followed by a liability convention in 1971.
It is state there that the launching state is liable for any damage caused by a space object.