I have never used LiveView, but I'd love to see the web moving towards this kind of interfaces. HTML over the wire is powerful! The deep integration between all the parts (Phoenix, LiveView, the HEEx engine...) is nice, and compile-time template validation looks really cool.
One thing that concerns me about doing everything over WebSockets, is that it seems you now need to keep a connection with every connected client, even if they are not doing any "highly interactive" actions. I think (in most cases) it would be more efficient to do a bunch of AJAX requests to receive those HTML chunks. Now that we have CDNs, HTTP/{2,3}, etc. the benefits of using WebSockets for everything seem less obvious.
This is what HTMX[0] does by default, and then you can use WebSockets[1] if you need it. Another great thing about HTMX is that the backend can be whatever you like, as long as you can handle HTTP requests and return HTML.
In any case, I like both approaches and I would love to see the web development ecosystem moving back to sending HTML over the wire, regardless of the framework.
[0] https://htmx.org/
[1] https://htmx.org/attributes/hx-ws/