With the help of someone else, I managed to figure this out.
Add this to your functions.php:
// Scheduled posts should update modified date when published
function update_modified_date_to_post_date( $post ) {
$updated_data = [
'ID' => $post->ID,
'post_modified' => $post->post_date,
'post_modified_gmt' => $post->post_date_gmt
];
wp_update_post( $updated_data );
}
add_action( 'future_to_publish', 'update_modified_date_to_post_date', 10, 1 );