Set post to draft after set period based on post_modified date

Don’t use global $post. Everything you need should be passed into the function in the $data and $postarr parameters.

Secondly, you have to return the $data in all cases. What you are doing now is effectively erasing data for everything but your profile post type.

You need to reorganize a bit:

function reset_post_date_wpse_121565($data,$postarr) {
  if ($data['post_type'] == 'profile'){
    $data['post_date'] = $data['post_modified'];
    $data['post_date_gmt'] = $data['post_modified_gmt'];
  } 
  return $data;
}
add_filter('wp_insert_post_data','reset_post_date_wpse_121565',99,2);

I have not tested that thoroughly but that is the right idea.

As for a better way…

Your expire_posts function is not going to be reliably portable as it uses a hard-coded database prefix. At the very least, rewrite that SQL so that the prefix is not hard-coded:

function expire_posts() {
  global $wpdb;
  $daystogo = "30";
  $sql = "UPDATE {$wpdb->posts} SET `post_status` = 'draft' WHERE `post_type` = 'profile' AND DATEDIFF(NOW(), `post_date`) > '$daystogo')";
  $wpdb->query($sql);
}
add_action('init', 'expire_posts');

I also just notices that your WHERE clause has a column that doesn’t exist. There is no the_date column in the $wpdb->posts table. It should be post_date.

Honestly, if you were to deal with the modified date only, you should be able to get this to work without ever altering the post date at all. I don’t think you need that second block of code at all.