The missing part to get it work for for cart and checkout pages (and also Orders and email notifications too) is a very simple trick:
add_action( 'woocommerce_before_calculate_totals', 'set_cart_item_sale_price', 20, 1 );
function set_cart_item_sale_price( $cart ) {
if ( is_admin() && ! defined( 'DOING_AJAX' ) )
return;
if ( did_action( 'woocommerce_before_calculate_totals' ) >= 2 )
return;
// Iterate through each cart item
foreach( $cart->get_cart() as $cart_item ) {
$price = $cart_item['data']->get_sale_price(); // get sale price
$cart_item['data']->set_price( $price ); // Set the sale price
}
}
Code goes in function.php file of your active child theme (active theme).
Tested and works.
So the code just set the product sale price as the product price in cart items and it works.
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