In general you're absolutely right but there are some parts of the City where there is some activity over the weekend. For example, the area around St Paul's and Cheapside usually have quite a few shops open and it's quite a pleasant place for a walk on a Sunday (and e.g. the Daunt's bookshop on Cheapside is very good and open both Sat and Sun).

It's interesting that it's quite the opposite just over the river (200m walk say) where places like Tate Modern and around London Bridge are usually very busy indeed over the weekend.