You’re already on the right track. Simply add conditional if 0
<?PHP
$cart_count = WC()->cart->get_cart_contents_count();
$cart_count = $cart_count ? "({$cart_count})" : ''; // if 0, become empty string
$cart_link = wc_get_cart_url();
$cart_button = "<a href="https://wordpress.stackexchange.com/questions/378487/{$cart_link}">Cart {$cart_count}</a>";
echo $cart_button;