Laravel: How to register an own implementation of `DatabaseManager` and `Connection`

Question

Due to debugging reasons (I am hunting down a very weird bug), I need to extend the Laravel classes Illuminate\Database\DatabaseManager, Illuminate\Database\Connection, and Illuminate\Database\Query\Builder.

Hence, I created the following three classes in my application.

namespace App\Database;

use Illuminate\Database\DatabaseManager as BaseDatabaseManager;

class DatabaseManager extends BaseDatabaseManager {
  public static bool $isDebugEnabled = false;

  // ...
  // Overwrites some methods of the base class which prints additional diagnostic information
  // if `self::$isDebugEnabled === true`.
  // ...
}

and

namespace App\Database;

use Illuminate\Database\Connection as BaseConnection;

class Connection extends BaseConnection {
  public static bool $isDebugEnabled = false;

  /**
   * Get a new query builder instance.
   *
   * This returns our own extended query builder with instrumentation.
   *
   * @return Query\Builder
   */
  public function query(): Query\Builder {
    return new Query\Builder(
      $this, $this->getQueryGrammar(), $this->getPostProcessor()
    );
  }


  // ...
  // Overwrites some methods of the base class which prints additional diagnostic information
  // if `self::$isDebugEnabled === true`.
  // ...
}

and

namespace App\Database\Query;

use Illuminate\Database\Query\Builder as BaseBuilder;

class Builder extends BaseBuilder {
  public static bool $isDebugEnabled = false;

  // ...
  // Overwrites some methods of the base class which prints additional diagnostic information
  // if `self::$isDebugEnabled === true`.
  // ...
}

How do I register my custom DatabaseManager and Connection with the Laravel Service Container instead of the original parent classes? The original classes are registered in \Illuminate\Foundation\Application::registerCoreContainerAliases() which is called by the constructor of the Application object, i.e. these classes are registered at a very early stage.

I thought that I could "re-register" my classes in \App\Providers\AppServiceProvider::register like this

namespace App\Providers;

class AppServiceProvider extends ServiceProvider
{
  /**
   * Register any application services.
   *
   * @return void
   */
  public function register() {
    // Overwrite core container services with our own classes
    // for debugging purposes
    // See \Illuminate\Foundation\Application::registerCoreContainerAliases()
    $this->app->alias('db', \App\Database\DatabaseManager::class);
    $this->app->alias('db.connection', \App\Database\Connection::class);
  }
}

but it did not work. The original classes are still being used. I used app->alias, because this is what registerCoreContainerAliases originally uses, too. But I also tried ->bind with no success.

After some debugging I found out that ->alias and ->bind do not overwrite a previously registered binding, but simply add a new binding to the end of the list. Hence, the previously registered services by the Laravel core application are still preferential.

What is the correct solution?