I can't tell what your description of LiveView is suggesting, I will need to read about it, sounds interesting. Is it like a codebase that is implicitly split on a back-end and a browser, without the need to write JavaScript?
LiveView is the haven for apps that are focusing on B2B and internal tools in organizations. Because it doesn't have to support offline usage, and cost effective since you don't have to have a frontend team.
The user's actions are sent to the Backend via WebSocket, and the Backend rerenders (parts of) the view and sends them to the user.
So "Welcome to $mysite, $username." would initially send the full rendering but if the data changes it would only send "Hacker News,lawik" over the wire. Somewhat simplified.
This covers it quite well: https://www.youtube.com/watch?v=8xJzHq8ru0M