Magento – Add to Cart is not working – Magento 2.2 – with Browser: Chrome v61.0

Well it's because the event you're trying to observe never gets dispatched by Magento. In Magento 2 controller_action_predispatch_skip_shoppingcart does not exist. I assume you wrote that event name by yourself assuming it had to be custom.

You need to replace that with controller_action_postdispatch_checkout_cart_add to observe the process after a product has been added to cart.

On top of that, you never call your afterAddToCart method in your observer, so only your execute() method will be called.

Finally, you have some Magento 1 code in there, so you need to replace:

    $response->setRedirect(Mage::getUrl("checkout/onepage"));
    Mage::getSingleton("checkout/session")->setNoCartRedirect(true);

With:

    $response->setRedirect($this->_urlManager->getUrl('checkout/onepage'));
    $this->_checkoutSession->setNoCartRedirect(true);

Where $this->_urlManager needs to be injected via Dependency Injection using the \Magento\Framework\UrlInterface class and $this->_checkoutSession using the \Magento\Checkout\Model\Session class.

Are you using apache? Make sure .htaccess file is been read or else try adding index.php to url.

Also make sure Php rewrite module is enabled.