Luckily, with Carina you don't need a "full" local Docker installation with a virtual machine, etc. You only need the Docker client, which is a standalone binary. I agree that being able to run a container from the website would be cool, and especially helpful for non-technical users, but here's what you can do today hopefully without too much trouble:
Checkout Getting Started on Carina, a tutorial which walks you through creating a cluster on the website and running containers from the command-line, without having a "full" local Docker daemon installation. It covers creating the cluster, downloading the credentials, connecting to your cluster with the Docker client, and running a few containers.
Thanks for the feedback!a