Also worth mentioning that free licenses like Creative Commons, MIT, and FOSS licenses like GPL are also all copyright schemes, as unintuitive as that might seem.
The removal of all copyright also means the removal of licenses that mandate free sharing and access. Two-way street, in other words.
Those liscenses, the GPL in particular, use copyright to fight copyright. The alternative to copyright is not necessarily a total free-for-all, you could legislate a replacement that works like the (A)GPL or CC BY(-SA). Require attribution and share-alike without restricting usage or distribution.
GPL uses copyright to ensure that software that makes use of GPL licensed code will be free (libre).
Even in a world without copyright protections, the printer story ( https://www.gnu.org/philosophy/rms-nyu-2001-transcript.txt ) would still have occurred, but there would be no way to ensure that it won't happen again with GPL licensed code.
Without copyright, open source code could still be locked up behind paywalls and compiled binaries. It is copyright that ensures that the law is there to force people to release their code.
The hypothetical legislation to ensure the GPL or CC-BY-SA works is copyright.
You could replace the license with a "contract" in the terms and agreements section to achieve the same effect? Contract law would apply, so I have no idea.