Is there any reason to register the copyright to your PhD dissertation with the US Copyright Office?
Registering a copyright with the US Copyright Office is a legal formality that establishes a public record of your claim to the work.
It is mainly useful if, in the future, you want to sue someone who infringes your copyright. Specifically, 


  
  Before an infringement suit may be filed in court, registration
  is necessary for works of U.S. origin.
  If made before or within five years of publication, registration
  will establish prima facie evidence in court of
  the validity of the copyright and of the facts stated in
  the certificate.
  If registration is made within three months after publication
  of the work or prior to an infringement of the work,
  statutory damages and attorney’s fees will be available to
  the copyright owner in court actions. Otherwise, only an
  award of actual damages and profits is available to the
  copyright owner.
  


These and some other similar benefits are described in this circular from the US Copyright Office.

Based on your post, you seem to have some misconceptions about how copyright works. Note that:


Even if you do not register the copyright, the work is still copyrighted. In the United States and many other countries, a creative work is protected by copyright automatically as soon as it is written down or otherwise placed in a fixed medium. 
Even if you do register the copyright, you can still make the work available under a permissive license, such as a Creative Commons license. See Is Creative Commons against copyright? for more information. For example, with CC-BY-NC you can choose to allow most non-commercial uses of your work as long as it is attributed to you, but still prohibit others from selling your thesis as an expensive book on Amazon.
You mentioned that you want to "let people use my work and cite me appropriately." Copyright does not prevent people from doing either of those things. Even if you reserve all rights under copyright, people are still free to use the ideas in your thesis. Copyright only protects your expression of your ideas, i.e. the words or images you used to express your ideas. So copyright prohibits people from copying large parts of your thesis verbatim into their own, or from posting copies of your thesis on another website (unless you choose to allow those things by assigning a permissive license to the work), but not from building on your work (and citing you). See Does copyright protect the author's creative ideas? for more information.
You wrote "a thesis should not be copyrighted simply because I think of it as an extended paper." The vast majority of papers are also copyrighted.