WordPress “Posts Page” isn’t showing template dropdown

Ben,

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Page Templates are only able to be defined in the Page post type (“Pages” in the WP-admin).

https://developer.wordpress.org/themes/template-files-section/page-template-files/#uses-for-page-templates

You can edit your theme (or create a child theme) to define how your post pages are displayed. Depending on your needs you could define an archive.php, category.php or index.php in your theme and then write the PHP/HTML necessary to display your posts the way you’d like. You can find more information on that here:
https://developer.wordpress.org/themes/template-files-section/page-template-files/#page-templates-within-the-template-hierarchy

Hope this help.