Same date is repeating on my custom ‘Most Recent Posts’ on sidebar

The same date is always displayed because of the use of echo get_the_date('F j, Y').
As you can read in the documentation:

The get_the_date template tag retrieves the date the current $post was written.

Change get_the_date('F j, Y') to get_the_date('F j, Y', $value->ID) and it should work.

Or without additional DB queries:

<div class="c-singlepost__sidebar__articles-item-date">
    <span>  <?php 
         $date = \DateTime::createFromFormat('Y-m-d H:i:s', $value->post_date);
         echo ($date !== FALSE) ? $date->format('F j, Y') : ''; 
    ?> </span>
</div>