What I’m trying to do is make a separate page on the site that lists all posts (not sorted by tag/category/year) in a single, easily-accessible portion of the site by just going to /archive (e.g. www.example.com/archive)
The correct approach is, indeed, to create a custom page template, so that the user can create a static page (titled, e.g. Archives), and then assign the custom page template to that static page. The custom page template content would then be accessible via www.example.com/archives.
(Note: I would avoid any assumptions that involve static pages being named the same as WordPress core-reserved slugs, such as archive.)
If I could, I’d rather make a theme that already has the feature built in without the user having to manually do the work to create the page.
Themes should not create user content out-of-the-box. Themes should only define how user-generated content is presented. The user should create all content.
Also, for some reason I never had a “template” dropdown section appear; go figure.
Create this file, in your Theme’s root directory: template-archives.php
Inside template-archives.php, add the following code:
/**
* Template Name: Archives
*/
You will now see the “Template” drop down (with “Default” and “Archives” listed) in the Page Attributes meta box on the edit page screen.
Once you have verified that you see the drop down, congratulations! WordPress recognizes your custom page template, and you are then free to populate it with whatever content you want.