Hi there, these are the lines (19-25) in content.php that are restricting the featured image to only certain post formats:
<?php if ( '' != get_the_post_thumbnail() && '' == $format ) : ?>
<a href="<?php the_permalink(); ?>" title="<?php echo esc_attr( sprintf( __( 'Permalink to %s', 'writr' ), the_title_attribute( 'echo=0' ) ) ); ?>" rel="<?php the_ID(); ?>">
<?php the_post_thumbnail( 'featured-image' ); ?>
</div><!-- .post-thumbnail -->
<?php endif; ?>
You can remove (or comment out) the if statement around the featured image code to remove the condition. The if statement is the first and last line above - lines 19 and 25. (Keep in mind that line numbers may differ if you've made any other edits to this file.)
I tested this on my end and it works to display the featured image on the main blog page no matter what post format is selected - audio, quote, image, etc.
As always, the best way to make changes to a theme is to use a child theme, so your tweaks won't be overwritten when updating the theme. Here are some guides in case you haven't made one before:
Let me know how it goes.