Remote Control no longer gives auto mode to sessions that cannot use it

Auto mode lets Claude Code act without asking for permission on each step. When Remote Control starts a child session (a session launched from another one), it decides which mode that session gets. It now checks whether auto mode is available first. If it is not, the child session is not granted auto mode, and Claude Code records the reason as auto mode being unavailable.

Before, the only case where auto mode was held back was when another mode had already been chosen.