I like the idea of adding the snippet you provided, as it eliminates a lot of redundancy.
So I tried both a and b (separately):
a) replacing <?php get_template_part('templates/content', 'page'); ?> with <?php get_template_part('templates/content', get_post_type() != 'post' ? get_post_type() : get_post_format() ); ?> and also creating the separate content files (i.e., content-page-products.php)
b) keeping my separate page files (i.e., page-products.php) and also corresponding content files (i.e., content-page-products.php).
I then cleared everything out of content-page-products.php and typed the word “Test”. In neither case was there any sort of change. The separated content files are saved within the /templates directory`, but it doesn’t look like this organization is working.