[LLDB] Use shared_ptr for m_current_private_state_thread (#188542)

Avoids manual memory management.

Uses `shared_ptr` instead of `unique_ptr` because we store references to
the current thread in a backup variable.

Simplifies the private thread `is_secondary` semantics by providing a
backup storage for the current thread instead of a boolean value with a
contract to manage the backup separately.

GitOrigin-RevId: b95b902bb7dd8875efe75a6955a8a4b4c7063d25