So, to answer my own question.
In the functions/backend code I made a mistake with this line:
'id' => $prefix . 'media_embed',
In a repeater field it it doesn’t need the prefix and as that is defined in the parent and so it should be:
'id' => 'media_embed',
And for the front end this line:
<?php $value = get_post_meta( get_the_ID(), 'media_embed', true ); echo wp_oembed_get( $value ); ?>
Was changed to:
<?php echo wp_oembed_get ($value['media_embed']); ?>
And that did the trick!