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>

priv/gettext/default.pot

@@ -2000,3 +2000,8 @@ msgstr ""
 #, elixir-autogen, elixir-format
 msgid "e.g. \"What genre is this?\", \"Tell me about the artists\""
 msgstr ""
+
+#: lib/music_library_web/components/record_chat.ex
+#, elixir-autogen, elixir-format
+msgid "Something went wrong. Please try again."
+msgstr ""

priv/gettext/en/LC_MESSAGES/default.po

@@ -2000,3 +2000,8 @@ msgstr ""
 #, elixir-autogen, elixir-format
 msgid "e.g. \"What genre is this?\", \"Tell me about the artists\""
 msgstr ""
+
+#: lib/music_library_web/components/record_chat.ex
+#, elixir-autogen, elixir-format
+msgid "Something went wrong. Please try again."
+msgstr ""