I have read up on Entity Systems in the articles that kind people provided in the comments, but I still had some doubts, so I asked another questionanother question.
First off, my definitions were wrong. Entities and Components are just dumb data holders, while systems provide all functionality.
I learned enough to cover most of my question here, so I'll answer it.
The Scene class I was talking about shouldn't be a system. It should, though, be a central manager that can hold all entities, facilitate messages, and maybe even manage systems. It can also function as a sort-of factory for entities, and I've decided to use it like that. It can take any type of an entity, but then it must feed that entity to an appropriate system (which, for performance reasons, shouldn't perform any type-checks, unless they're bitwise).
I shouldn't use any OOP while implementing an ES, suggests Adam, but I find no reason not to have objects with methods for both entities and components, not just dumb data holders.
The Renderer can simply be implemented as a system. It would maintain a list of drawable objects and call their render component's draw() method each 16ms.