Are your reverse proxy/ websockets working correctly? When I first started the UI without adding the routes for the backend, the login button would just time out exactly like that.
See if there are any network errors in your browser's developer tools, and see if there are any logs in your backend container.
Are you running an nginx container as well? The nginx config is pretty complicated and crucial, so I would recommend pointing caddy to the nginx container and let it handle all the different routing.
The documentation is not currently in the greatest state, from what I saw. There is literally nothing other than the default config for how to setup email / SMTP.