> it would be caught fairly quickly
Noticing something and reacting to it are very different things. Signal could fairly trivially grab all historical data for all online users within a fairly limited window. However it would be a one off event so the value proposition of such an act is dubious.
> fairly trivially
Show your working otherwise this is utterly spurious.
What is complicated about having the local client upload its database to a remote endpoint? It's literally opening a network connection and proceeding to write out a database dump to it.
Anyway the difficulty of the task itself is traditionally taken to be irrelevant when performing cryptographic threat analysis. The question is about what is and is not mathematically impossible for an adversary to do.