Hmm, That was very helpful. Although my nav menu is still acting like there is a search there (it’s pushing my nav menu items into a new row, even though it appears to have plenty of space left). How do I take the search removing to the next level?
Edit: I got what I wanted by just deleting the search div:
<div class="only-search<?php if ( ! empty( $header_image ) ) : ?> with-image<?php endif; ?>">
<!-- <?php get_search_form(); ?> -->
</div>
