Venn diagrams make for very iconic representations of their universes of discourse. That is one of the main sources of their intuitive utility and also the main source of their logical limitations — they begin to exceed our human capacity for visualization once we climb to four or five circles (Boolean variables) or so.
Peirce’s logical graphs at the Alpha level (propositional calculus) are somewhat iconic but far less so than Venn diagrams. They are more properly regarded as symbolic representations, in a way that exceeds the logical capacities of icons. That is the source of their considerably greater power as a symbolic calculus.