Video controls don't show up on full width slider

but your video is not a self hosted one?

Even the div.custom-mute isn’t inserted – so please try with document ready function:
( i can create the div via developer tools console – so the selectors are correct )

function unmute_mute_background_video(){
?>
<script type="text/javascript">
(function($) { 
$(document).ready(function(){  
    $('.mute-button .avia_video').closest('.avia-slide-wrap').prepend('<div class="custom-mute"></div>');
    $('.mute-button .avia_video').on('av-mediajs-loaded', function() {   
      var video = $(this).mediaelementplayer();
      video[0].setMuted(true);
      $('.avia-slide-wrap').on('click', '.custom-mute', function() {
        if(video[0].muted === true) {
          video[0].setMuted(false);
          video[0].setVolume(0.5);
        } else {
          video[0].setMuted(true);
        }   
      });
    });
    
    $('.avia-slide-wrap').on('click', '.custom-mute', function() {
      $(this).toggleClass("icon-sound-on");
    });
});
})(jQuery);
</script>
<?php
}
add_action('wp_footer', 'unmute_mute_background_video');

i answered you per mail – i do not know why.
it seems that everything is done on your site. But the div.custom-mute isn’t inserted.
You can see on the test-page link i send you via e-mail that even with your video ( but downsized to 26MB) it works.
( it works on my side here even without document ready or window load function. )
And even if i got two videos on the slider – i do not need to have the custom-mute div generated on an each function. – it works the way above – all video slides will have that buttons.

on that add_action line can you please do this :

add_action('wp_footer', 'unmute_mute_background_video', 999);

so that there is:

function unmute_mute_background_video_in_slider(){  
?>
<script type="text/javascript">
(function($) { 
    $('.mute-button .avia_video').closest('.avia-slide-wrap').prepend('<div class="custom-mute"></div>');
    $('.mute-button .avia_video').on('av-mediajs-loaded', function() {   
      var video = $(this).mediaelementplayer();
      video[0].setMuted(true);
      $('.avia-slide-wrap').on('click', '.custom-mute', function() {
        if(video[0].muted === true) {
          video[0].setMuted(false);
          video[0].setVolume(0.5);
        } else {
          video[0].setMuted(true);
        }   
      });
    });
    
    $('.avia-slide-wrap').on('click', '.custom-mute', function() {
      $(this).toggleClass("icon-sound-on");
    });
})(jQuery);
</script>
<?php
}
add_action('wp_footer', 'unmute_mute_background_video_in_slider', 999);

now without function ready