The thing that trips most people up is that the parameters to the web cli (--backend elasticsearch://blah:9300/blah) do NOT set the output destination.
The web app is its own process with its own args. It knows nothing about the agent config file.
Remember that you could simply use the same jar like so:
java -jar logstash.jar web --backend elasticsearch://elasticsearch:9300/clustername
and run the web interface without the agent process.