How do I add custom HTML to the content of an archive page’s posts?

Try changing the priority, there could be other filters being applied afterwards, i.e:

add_filter( 'the_content', 'eri_course_archive_content_filter', 999 );

If that doesn’t work, check that the_content() is actually being called in your archive template, because for example:
echo get_the_content(); will not apply any filters set on the_content