The visibility is set to hidden, but the media-element-player JS and CSS code makes it show up again with the proper player controls and so forth. This is a bug fix for a potential race condition.
More info here: https://core.trac.wordpress.org/changeset/28182
Specifically, the reason your players were not showing properly was not really because of the visibility hidden, but because that page at http://www.ktbb.com/audio/ is somewhat incomplete.
None of the mediaelementplayer JS or CSS code is showing up on that page. In fact, none of the normal WordPress header elements are. So, I suspect the problem is that your theme lacks the correct
wp_head() and possibly the
wp_footer() calls in the header.php and footer.php files, respectively.
Before the ending
</head>, your page needs to have this code:
<?php wp_head(); ?>
And before the ending
</body>, this code:
<?php wp_footer(); ?>
<audio> tag, the media-element-player code included in WordPress is what makes that audio tag cross-browser.