My concern isn't really "does the program keep running?", it's "does the program keep running correctly?".
Erlang/Elixir do make this pretty easy to manage, including the scenario where the process does recover by reverting back to a known good state. It won't do it for you automatically, but it exposes enough surface area to make problems like that solvable without reaching for a lot of extra tools - it's built into the runtime.
Yes, which is why Go's outright crashing also makes sense to me...both Go and Erlang's behavior seem conceptually the same, with some architectural tradeoffs. It's not really that different for a process to die and restart. If some shared resource reaches an undefined state, then you have to kill everything and reset your state anyway. I suppose Go's behavior lends itself better to "microservices", whereas Erlang's behavior is better suited for "monolith" processes that do a lot of different things.
IMO either of these are better than Java's default behavior of silently swallowing the exception and allowing the thread to quietly die.
If you're in a coordinated dance with another process you link to that process. If a process you're linked to crashes then you crash too. There's no way to block yourself in Erlang such that you can't be told to crash.
After you crash your supervisor might restart you, if that's what you configured. Or you might give up on your specific task.