A reason you have limited ability to search "copyrights" is because there is usually limited or no reason to search them.
You don't need to search before registering your own copyright, but there are other reasons to search copyright office records. In particular, you may want to identify the copyright owner of an existing work for licensing/permission purposes. Additionally, if you are trying to identify whether a work has fallen into the public domain, details such as renewals, dates of publication, and registration details can be important, particularly for older works that date back to a time when copyright protection was not automatic.
Its also worth noting that while copyright registration can helo you in the event you must enforce your rights in a court of law, copyright protection tends to be automatic.
If you are creating works in the US, you cannot sue for infringement in a US court until you register (or at least file for registration). You can register after the infringement occurs, but that may result in the loss of some remedies for infringement.
Determining whether or not a film is still under copyright protection involves not a copyright search, but rather an understanding of copyright law. If you understand copyright law, you can then determine whether or not a film is in the public domain.
Figuring out whether a work is in the public domain is not as easy as you suggest. To apply the law, you often need to know details that are available in Copyright Office records and often no place else.