Now, I know Erlang much better than Elixir, and I worked with Cowboy before and I needed to make this app quickly, which resulted in me not spending too much time on learning Phoenix. Between controllers, routes, views and channels I got an impression that Phoenix has too many moving parts and that it would take me too much time to fully understand what's going on. Especially because I really didn't need most of these, just a static file server and a single WebSocket.
However, I see Phoenix potential for more complex projects, where investing the time to learn it is going to be worth it. It looks like Phoenix provides an awful lot of conveniences and makes a project much better structured than my "a couple of files in a single directory" approach.
I guess what I want to say is that I almost used Phoenix this time and that I would probably use it if it had better docs - especially a solid tutorial(s) for use cases similar to mine. And that, while I didn't use it this time around, I'm certainly going to keep an eye on it and consider it next time I have to write something similar. It looks very promising and - like Elixir itself - very interesting, I hope for it to only grow in the future :)