How to output custom post type title on custom page with category next to it?

$leaderQuery->the_post() should only be called once, right after the loop starts:

while ( $leaderQuery->have_posts() ) {
    $leaderQuery->the_post(); // call it just once
    ...
}

I could also see you’re not correctly retrieving the ID of the current post in the loop, so the correct way is use get_the_ID().

$postCategories = get_the_category( get_the_ID() );           // like this
$postCategories = get_the_category($leaderQuery->the_post()); // not this

But within a loop, you can simply call get_the_category() like that, i.e. without any parameters.

And note that get_the_category() returns an array of term objects, so if you just want to get the name of the first category in that array, then you can do:

// Get all categories (term objects) assigned to the current post.
$postCategories = get_the_category();

// Get the first category's name.
$first_cat_name = ( ! empty( $postCategories ) ) ? $postCategories[0]->name : '';

But if you want to display all the categories, then you can simply use the_category().

And note that for custom taxonomies, you would:

Use get_the_terms() and not get_the_category().
Use the_terms() and not the_category().

Update

Actually, in your WP query’s args, that cat_slug is not a valid parameter for WP_Query. Instead, it should be category_name, but remember that it expects a slug and not the category name, e.g. my-category and not My Category.

Also, you could use something like so to display a specific category name only if the current post is in that category:

if ( in_category( 'Leadership' ) ) {
    echo 'In the Leadership category';
}

// Or after closing the PHP tag ( ?> ):
//<?php echo in_category( 'Leadership' ) ? 'In the Leadership category' : ''; ?>

Update

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