How can users write a bio on their profile page?
Re: the second/existing option, if you have a custom module already, could do a form_alter() on the node form to remove the title field.  Or, you could look into the auto_nodetitle module that does this for you:


  "auto_nodetitle" is a small and efficient module that allows hiding of
  the content title field in the form. To prevent empty content title
  fields one can configure it to generate the title by a given pattern.
  
  When the token module is installed it's possible to use various node
  data for the auto generated title - e.g. use the text of a CCK field
  (since 5.x).
  
  Advanced users can also provide some PHP code, that is used for
  automatically generating an appropriate title.


If instead you wanted to stay/go back to the first option, you could create a menu link/path like user/%/edit/bio that presents the user_profile_form that has been form_alter()ed to show only the bio field based upon the fact that it was called from this new path.