What is the most drupal-like way to bring the title of a single node page into the rest of the content?
It is valid to do it the way you are doing it and it would also be valid to not print the title in the page tpl for certain nodes and instead print it in the node template.

It's subjective as to which is preferable (if one is even better than the other) and the preferred solution may change based on your specific requirements for a given project.

One thing to consider is anything that usually prints out between the title and the content in your page template.
Will moving the title into the content (in this case the content is essentially the node template) mean these nodes will be inconsistent to the rest of the site (for example, will messages/tabs/help/etc. be in a different place on these nodes)? 
If they were to be different would it matter to you?

Yes, you can still use hide($content['field_doc_cover']); in your node template in this case.

There are also alternatives depending on your site and requirements. For example the Panels module allows you to assign specific fields to different regions and there are a number of ways to get a block to show parts of a node so that you can assign that to a different region using Drupal core or the Context module or similar.