Editing display of submission results
Is there configuration I'm not seeing to adjust the submission edit screen


No, there is no config screen to customize this "edit" screen. 


  the client doesn't want to see all of the fields and
  submission-related information (UUID, URI, etc.)


You can use hook_form_alter() in a custom module and remove the submission information section.


  and wants to see all of the form submission fields filled out here as
  well.


They should be already be filled out, unless the user left them empty. Make these fields required if they should be filled out.


  I also see there's a Webform Views Integration module, but that
  looks more like what I'd want if I wanted to put a view within a form.


In correct, this allows you to display the webform results in a View. This is exactly what you need, well if you want to make an alternate page that has that "edit" info. 

Also, you could alter the Webform's edit route so it uses your Webform View instead, but you would lose the edit ability tho. Since the View just displays what was entered and is not a form.