My thought was that the CSD would rotate on different flights between the cabins, if they were in First for that flight then the First purser would move to Club and be responsible for the Cabin checks under the supervision of the CSD (and Captain!).
When I said "helped" I meant helped to achieve high standards of service by the CSD, who you imply has raised the standards in Club. As a traveller in First, there is sometimes an inconsistent service, the CSD being involved throughout the cabins would quite possibly raise the standard everywhere. (Though I accept you can't be everywhere and Prem pax may be miffed)
I don't see a problem with the CSD working from Traveller from a FD perspective, after all I'm usually told that "******* will be taking over during rest and will be at Door 4L"
P.S. It's nice to be looking forward for a change
