Hi,
You can create a custom HTML module instance to add an img tag for your cart icon inside an a tag with its src attribute set to the URL of the checkout. That way, when the customer would click on the cart icon, he would go to the checkout.
Alternatively, you could enable back the cart and the notify plugin and instead configure the notify plugin to display a centered popup notification. That way, after the add to cart, the customer would have an easy to see notification popup in the middle of the screen with a "proceed to checkout" button in order to directly access the checkout.