Echoing what some others have said to some degree, but adding a few points (hopefully! 
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You make the decision per route in your site whether (and how) to use LiveView. I have a couple significant apps that mix and match as makes sense. I’ve even transitioned some ‘regular’ pages to LiveViews, and some LiveViews to ‘regular’ pages as requirements became clearer or evolved. It’s really flexible like that, and not a one-way-OR-the-other choice.
You also said you might just go with “plain (web)sockets”, well .. that’s what LiveView is, only with a well-defined lifecycle and efficient handling of DOM changes. So it takes a lot of the stuff-one-can-get-wrong off our shoulders.
That said, sometimes I’ve had to use the pushEvent /push_event functions to pass messages over the websocket that the LiveView uses in order to integrate with more complex / dynamic client-side JS libraries (such as Sortable and OpenLayers). That’s the same as “over a plain socket” at the end of the day, but only as needed (LiveView can handle the rest..)
So the choice between LiveView vs “just messages over the socket” also isn’t binary: you can do both at the same time. I certainly do!