Prev/Next Links Broken on Static Front Page Pagination

previous_posts_link() echos the result of get_previous_posts_link() which checks the paged query variable, not the page query variable used on static pages. You will have to write your own code to display the link on a static page.

I didn’t test it, but you might try fooling WordPress:

function homepage_test() {
    global $paged;

Now your changes to the $paged variable in your code will be seen by get_previous_posts_link(). I don’t know what else it might affect or if the link will be correct.