As soon as I found out that Unreal Engine 4 is now free, I decided to get it in order to tinker with it.
So I clicked on the blue "Get Unreal" button, signed up, and downloaded (and installed) the MSI installer (working on Windows here).
After the installation completed, the "Epic Games Launcher" launched, asking me to log in. Once logged in, I selected the "Unreal Engine" tab and was greeted by a page with several tutorials and demo projects for download, while the left side offered an orange button saying "No engine installed" and no option to click on it to install one.
After some time, I found this Unreal documentation explaining how to install the engine. According to it, the engine should start downloading the moment I log in to the launcher, which didn't happen.
What am I doing wrong? How can I install the Unreal Engine?