Yeah you can definitely pull in Alpine for UI updates that don't require any server side interaction, but then your app's logic is fragmented between different frameworks (Alpine and LiveView).
It's definitely doable, and I know the PETAL stack (Phoenix Elixir Tailwind Alpine) is super popular, but it's nice (and perhaps less error prone) to have all of your UI logic handled by a single framework. This consolidation of UI code under a single framework was one of the killer features of SPAs initially. Prior to that lots of sites were built with a server side templating framework with jquery sprinkled in for client side UI updates.
It's definitely doable, and I know the PETAL stack (Phoenix Elixir Tailwind Alpine) is super popular, but it's nice (and perhaps less error prone) to have all of your UI logic handled by a single framework. This consolidation of UI code under a single framework was one of the killer features of SPAs initially. Prior to that lots of sites were built with a server side templating framework with jquery sprinkled in for client side UI updates.