To achieve this, create a basic module and below code in InstallSchema.php file.

Change the vendor namespace and module name as per your requirement.

<?php
namespace MagemonkeysDiscontinueSetup;

use MagentoEavSetupEavSetup;
use MagentoEavSetupEavSetupFactory;
use MagentoFrameworkSetupInstallDataInterface;
use MagentoFrameworkSetupModuleContextInterface;
use MagentoFrameworkSetupModuleDataSetupInterface;

class InstallData implements InstallDataInterface
{
    private $eavSetupFactory;

    public function __construct(EavSetupFactory $eavSetupFactory)
    {
        $this->eavSetupFactory = $eavSetupFactory;
    }

    public function install(
        ModuleDataSetupInterface $setup,
        ModuleContextInterface $context
    )
    {
        /** @var EavSetup $eavSetup */
        $eavSetup = $this->eavSetupFactory->create(['setup' => $setup]);

        /**
         * Add attributes to the eav/attribute
         */
           $eavSetup->addAttribute(
            MagentoCatalogModelProduct::ENTITY,
            'discontinue',
            [
                'group' => 'Discontinue Product Information',
                'type' => 'int',
                'backend' => '',
                'frontend' => '',
                'label' => 'Discontinue',
                'input' => 'boolean',
                'class' => '',
                'source' => 'MagentoEavModelEntityAttributeSourceBoolean',
                'global' => MagentoCatalogModelResourceModelEavAttribute::SCOPE_GLOBAL,
                'visible' => true,
                'required' => false,
                'user_defined' => false,
                'default' => false,
                'searchable' => false,
                'filterable' => false,
                'comparable' => false,
                'visible_on_front' => false,
                'used_in_product_listing' => true,
                'unique' => false
            ]
        );

           /**
         * Add attributes to the eav/attribute
         */
           $eavSetup->addAttribute(
            MagentoCatalogModelProduct::ENTITY,
            'alternate_product_url',
            [
                 'group' => 'Discontinue Product Information',
                 'type' => 'text',
                 'backend' => '',
                 'frontend' => '',
                 'label' => 'Alternate Product URL',
                 'input' => 'text',
                 'class' => '',
                 'source' => '',
                 'global' => MagentoEavModelEntityAttributeScopedAttributeInterface::SCOPE_GLOBAL,
                 'visible' => false,
                 'required' => false,
                 'user_defined' => false,
                 'default' => '',
                 'searchable' => false,
                 'filterable' => false,
                 'comparable' => false,
                 'visible_on_front' => true,
                 'used_in_product_listing' => true,
                 'unique' => false,
                 'apply_to' => ''
            ]
        );
           $eavSetup->addAttribute(
            MagentoCatalogModelProduct::ENTITY,
            'alternate_product_sku',
            [
                 'group' => 'Discontinue Product Information',
                 'type' => 'text',
                 'backend' => '',
                 'frontend' => '',
                 'label' => 'Alternate Product SKU',
                 'input' => 'text',
                 'class' => '',
                 'source' => '',
                 'global' => MagentoEavModelEntityAttributeScopedAttributeInterface::SCOPE_GLOBAL,
                 'visible' => true,
                 'required' => false,
                 'user_defined' => false,
                 'default' => '',
                 'searchable' => false,
                 'filterable' => false,
                 'comparable' => false,
                 'visible_on_front' => true,
                 'used_in_product_listing' => true,
                 'unique' => false,
                 'apply_to' => ''
            ]
        );
    }
}

Create a Observer with catalog_product_save _before in adminhtml/events.xml

<?xml version="1.0"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:Event/etc/events.xsd">
    <event name="catalog_product_save_before">
        <observer name="discontinue_product_save_after" instance="MagemonkeysDiscontinueObserverProductsavebefore" />
    </event>
</config>

Add Observer file and place the below code in it.

<?php

namespace MagemonkeysDiscontinueObserver;

use MagentoFrameworkEventObserverInterface;
use MagentoCatalogApiProductRepositoryInterface;
use MagentoFrameworkUrl;
use MagentoFrameworkExceptionNoSuchEntityException;

class Productsavebefore implements ObserverInterface
{    

    /**
     * @var ProductRepositoryInterface
     */
    protected $productRepository;

    protected $_url;
 
    /**
     * @param ProductRepositoryInterface $productRepository
     */
    public function __construct(
        ProductRepositoryInterface $productRepository,
        Url $url
    ) {
        $this->productRepository = $productRepository;
        $this->_url = $url;
    }

    public function execute(MagentoFrameworkEventObserver $observer)
    {
        $_product = $observer->getProduct();  // you will get product object

        $sku = $_product->getAlternateProductSku();

        if($sku != '')
        {
            try
            {
                $alternateProduct = $this->productRepository->get($sku);
            }
            catch (NoSuchEntityException $noSuchEntityException) {
                $productUrl = null;
            }

            if($alternateProduct)
            {

                $product = $this->productRepository->getById($alternateProduct->getId(), false);
                //$productUrl = $product->setStoreId(1)->getUrlModel()->getUrlInStore($product, ['_escape' => true]);
                $productUrl = $product->getUrlKey();
                $_product->setData('alternate_product_url', $productUrl);
                
            }
        }
        else
        {
            $productUrl = null;
            $_product->setData('alternate_product_url', $productUrl);
        }

    }   
}

Create a plugin with aroundGetProduyctCollection to remove the discontinue product from whole site. Please add the below code in that plugin.

<?php

namespace MagemonkeysDiscontinuePluginMagentoCatalogModel;

class Layer
{

    public function aroundGetProductCollection(
        MagentoCatalogModelLayer $subject,
        Closure $proceed
    ) {
        $collection = $proceed();
        $collection->addAttributeToSelect('*')->addAttributeToFilter(array(
            array(
                'attribute' => 'discontinue',
                'null' => true),
            array(
                'attribute' => 'discontinue',
                'eq' => '0')
        ));
        return $collection;
    }


}