No, it really is the author's responsibility and publishers make sure to protect themselves.
All this below is from https://ochsnerjournal.wordpress.com/2015/02/05/copyright-infringement-who-is-responsible-author-or-publisher/
To strengthen their legal position, many publishers include warranty and indemnity clauses in the copyright transfer agreement. Usually, by agreeing to a warranty clause, the author guarantees that his manuscript contains only material that is original or in the public domain and free from any copyright. Often the author also promises to indemnify the publisher if the manuscript becomes a legal liability; an indemnity clause makes the author financially liable if he breaks the warranty clause. Here is an example of an indemnity clause from an academic publisher:
If a publisher is sued for copyright infringement and loses, the publisher can then sue the author for breach of contract and potentially recover damages awarded in the original lawsuit. The existence of warranty and indemnity clauses means that even though publishers are sued for copyright infringement more often than authors, authors are not free from liability. The only way that authors and publishers can avoid copyright litigation is by avoiding copyright infringement in an active, collaborative effort.