Does not seem to work when database is down

Yes, can confirm as well. Pasting full output below:

Additional uncaught exception thrown while handling exception.
Original
PDOException: SQLSTATE[HY000] [2002] Connection refused in Drupal\Component\DependencyInjection\PhpArrayContainer->createService() (line 79 of /code/web/core/lib/Drupal/Component/DependencyInjection/PhpArrayContainer.php).

Drupal\Component\DependencyInjection\PhpArrayContainer->createService() (Line: 177)
Drupal\Component\DependencyInjection\Container->get() (Line: 212)
Drupal\Component\DependencyInjection\PhpArrayContainer->resolveServicesAndParameters() (Line: 62)
Drupal\Component\DependencyInjection\PhpArrayContainer->createService() (Line: 177)
Drupal\Component\DependencyInjection\Container->get() (Line: 565)
Drupal\Core\DrupalKernel->getCachedContainerDefinition() (Line: 942)
Drupal\Core\DrupalKernel->initializeContainer() (Line: 487)
Drupal\Core\DrupalKernel->boot() (Line: 716)
Drupal\Core\DrupalKernel->handle() (Line: 19)
Additional
Error: Class "Drupal\error_page\ErrorPagePhpErrorLogger" not found in Drupal\error_page\ErrorPageErrorHandler::logError() (line 179 of /code/web/modules/contrib/error_page/src/ErrorPageErrorHandler.php).

Drupal\error_page\ErrorPageErrorHandler::logError() (Line: 102)
Drupal\error_page\ErrorPageErrorHandler::handleException()

Some exceptions are occurring very early, before Kernel gets a chance to initialize. Such a case is the database connection error. In that circumstance, the extension classes are not yet added to the class autoloader, making it impossible to load the classes.

We had to make the whole code more robust.

Let's switch the error & exception handlers to procedural code. Let's keep the current classes as deprecated to ensure BC.

IMPORTANT: Please check carefully https://www.drupal.org/node/3514863 → before testing because there are some changes needed for settings.php, even it should work with the current settings.

To test, just set an invalid MySQL password