I am trying to search a custom post type via title using a form and query variables, what I am struggling with is getting the results to update properly. Currently, when the page loads it shows all results as expected, but when I input something into the search field and search, it shows all the results instead of the expected one, the url does update with the query var as well.

archive-researcher.php

    <section>
      <form id="researcher" method="GET" action="<?php echo get_post_type_archive_link('researcher'); ?>">
        <div>
          <input type="text" name="r_name" placeholder="Search">
        </div>
        <div>
          <button>Search</button>
          <a href="<?php echo get_post_type_archive_link('researcher'); ?>">Reset</a>
        </div>
      </form>
      <div class="row">
        <?php if (have_posts()): ?>
        <?php while (have_posts()): the_post(); ?>
        <div class="col-12 col-lg-4 d-flex">
          <?php get_template_part('template/card', 'researcher'); ?>
        </div>
        <?php endwhile; ?>
        <?php else: ?>
        <div class="col-12">
          <p>No researchers found.</p>
        </div>
        <?php endif; ?>
      </div>
      <?php the_posts_pagination(); ?>
    </section>
  </main>
</div>


<?php get_footer(); ?>

functions.php

function researcher_archive($query)
{
  // only run this query if we're on the researcher archive page and not on the admin side
  if (
    $query->is_archive("researcher") &&
    $query->is_main_query() &&
    !is_admin()
  ) {
    // get query vars from url.
    // https://codex.wordpress.org/Function_Reference/get_query_var#Examples
    $r_name = get_query_var("r_name", false);

    
      $query->set("s", $r_name);
    



    $query->set('posts_per_page', 12);
    $query->set('orderby', 'wpse_last_word');
    $query->set('order', 'ASC');

    
  }
}
add_action("pre_get_posts", "researcher_archive");