Supervisors are critical to the application’s reliability, so it is recommended to always use OTP’s standard (proven, simple, robust) supervisors rather than roll your own.

If some part of your application needs to be notified when processes terminate, add a GenServer that uses Process.monitor/1 to track other processes. This is orthogonal to the supervision hierarchy and therefore any issues in your monitoring code will not impact supervision.

I am pretty late here, but I’m afraid one can not overwrite Supervisors handle_info/2, as well as handle_call and handle_cast, because while Supervisor does not define those callbacks.

defmodule Sup do
  use Supervisor

  def handle_info(any, state) do
    IO.inspect any
  end
end

# try messaging
send(sup_pid, :hello)

Trying the above does not work, it will say something like
01:52:12.737 [error] Supervisor received unexpected message: :hello

The handle_info handle_call the Supervisor can receive seems to be defined here, and Elixir is just calling that inside def which_children, do: ....

Maybe you can overwrite handle_info/2 in your supervisor module?

require Logger

def handle_info({:EXIT, pid, reason}, state) do
  Logger.info("A child process died: #{reason}")
  case restart_child(pid, reason, state) do # it's a private function ... that's a problem
    {:ok, state} ->
	  {:noreply, state}
    {:shutdown, state} ->
	  {:stop, :shutdown, state}
  end
end

def handle_info(msg, state) do
  Logger:error("Supervisor received unexpected message: #{inspect(msg)}")
  {:noreply, state}
end

It is my understanding that SASL Error Logging will capture the exit reason as part of the Supervisor Report. So it’s a matter of enabling SASL logging and capturing the log with an appropriate backend.