Fix review blockers in record chat

- Add Task.Supervisor to app supervision tree
- Use tagged tuples instead of rescue for error handling
- Pass embedding_text from parent LiveView as prop
- Remove dead Completion struct fields
- Extract socket assigns before Task closure
- Log errors server-side, show generic user message
- Make build_system_prompt private, remove thin wrapper
- Extract do_send_message for retry reuse

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

lib/music_library/application.ex

@@ -23,6 +23,7 @@ defmodule MusicLibrary.Application do
       {Oban, Application.fetch_env!(:music_library, Oban)},
       {Ecto.Migrator,
        repos: Application.fetch_env!(:music_library, :ecto_repos), skip: skip_migrations?()},
+      {Task.Supervisor, name: MusicLibrary.TaskSupervisor},
       {Phoenix.PubSub, name: MusicLibrary.PubSub},
       {LastFm.Supervisor, LastFm.Config.resolve(:music_library)},
       # Start a worker by calling: MusicLibrary.Worker.start_link(arg)