A (big) carrot and (small) stick approach may work.

The stick would be a simple requirement that copyright of older works must be explicitly registered in a publicly accessible database. While hardly onerous on a business intending to disseminate for profit, this would ensure that only documents with a known owner - that actually values them - would be restricted. Presumably "Steamboat Willie" would fall into this category, but the vast bulk of works of unknown history (or considered to have no value) would automatically become public domain.

The carrot would be a tax incentive to place copyrighted works explicitly into the public domain, such as archive.org, in much the same way as paintings are donated to museums today. This should be attractive to publishers who hold copyrighted works with some value, but not enough to make it worth the effort to publish.