Behavior on published content.

I needed this functionality as well. I modified the functionality on PreviewLinkController::preview() so that it:
1. checks entity->status->value to see if it's published
2. if published, calls the sessiontokencontroller and removes the tokens (to avoid redirect loops)
3. redirects to the new entity URL.

I'm pasting in the patch file while I figure out how to submit a patch - it's been a while so I need to remember how to do it.

diff --git a/preview_link.services.yml b/preview_link.services.yml
index 5f11fab..af709ef 100644
--- a/preview_link.services.yml
+++ b/preview_link.services.yml
@@ -51,6 +51,22 @@ services:
     arguments: ['@entity_type.manager']
     tags:
       - { name: event_subscriber }
+  preview_link.session_token_controller:
+    class: Drupal\preview_link\Controller\PreviewLinkSessionTokenController
+    arguments:
+      - '@string_translation'
+      - '@messenger'
+      - '@tempstore.private'
+  preview_link.controller:
+    class: Drupal\preview_link\Controller\PreviewLinkController
+    arguments:
+      - '@entity_type.manager'
+      - '@tempstore.private'
+      - '@config.factory'
+      - '@preview_link.message'
+      - '@messenger'
+      - '@preview_link.hook_helper'
+      - '@preview_link.session_token_controller'
   logger.channel.preview_link:
     parent: logger.channel_base
     arguments: [ 'preview_link' ]
diff --git a/src/Controller/PreviewLinkController.php b/src/Controller/PreviewLinkController.php
index d4cf5f3..8acd37c 100644
--- a/src/Controller/PreviewLinkController.php
+++ b/src/Controller/PreviewLinkController.php
@@ -15,11 +15,15 @@ use Drupal\Core\TempStore\PrivateTempStoreFactory;
 use Drupal\preview_link\PreviewLinkHookHelper;
 use Drupal\preview_link\PreviewLinkMessageInterface;
 use Symfony\Component\DependencyInjection\ContainerInterface;
+use Symfony\Component\HttpFoundation\RedirectResponse;
+use Drupal\preview_link\Controller\PreviewLinkSessionTokenController;
+
 
 /**
  * Preview link controller to view any entity.
  */
 class PreviewLinkController extends ControllerBase {
+  protected PreviewLinkSessionTokenController $sessionTokenController;
 
   /**
    * PreviewLinkController constructor.
@@ -44,10 +48,12 @@ class PreviewLinkController extends ControllerBase {
     protected PreviewLinkMessageInterface $previewLinkMessages,
     MessengerInterface $messenger,
     protected PreviewLinkHookHelper $hookHelper,
+    PreviewLinkSessionTokenController $sessionTokenController
   ) {
     $this->entityTypeManager = $entityTypeManager;
     $this->configFactory = $configFactory;
     $this->messenger = $messenger;
+    $this->sessionTokenController = $sessionTokenController;
   }
 
   /**
@@ -60,7 +66,8 @@ class PreviewLinkController extends ControllerBase {
       $container->get('config.factory'),
       $container->get('preview_link.message'),
       $container->get('messenger'),
-      $container->get('preview_link.hook_helper')
+      $container->get('preview_link.hook_helper'),
+      $container->get('preview_link.session_token_controller')
     );
   }
 
@@ -75,12 +82,18 @@ class PreviewLinkController extends ControllerBase {
    * @return array
    *   A render array for previewing the entity.
    */
-  public function preview(RouteMatchInterface $routeMatch, string $preview_token): array {
+  public function preview(RouteMatchInterface $routeMatch, string $preview_token): array|RedirectResponse {
     // Accessing the controller will bind the Preview Link token to the session.
     $this->claimToken($preview_token);
 
     $entity = $this->resolveEntity($routeMatch);
-
+    // Is the entity published?
+    $is_published = $entity->status->value;
+    if ($is_published && $routeMatch->getRouteName() !== $entity->getEntityType()->getLinkTemplate('canonical')) {
+      // Redirect to the published node.
+      $this->sessionTokenController->removeTokens();
+      return new RedirectResponse($entity->toUrl()->toString());
+    }
     $config = $this->configFactory->get('preview_link.settings');
     if (in_array($config->get('display_message'), ['always'], TRUE)) {
       // Reset static cache so our hook_entity_access is always re-evaluated.

Figured it out... pushed to the fork: https://git.drupalcode.org/issue/preview_link-3273377/-/compare/2.1.x......