JPL's Horizons is what I use for the starting positions and velocities in Universe Sandbox.
The interface is not user friendly... you send them an email and get back an email for each body you're requesting data for. 400 bodies? 400 emails.
The first thing that comes to mind is some kind of Gantt chart. Visualization is always something I've wanted to work on/improve for SimPy models, usually only wind up visualizing other outputs. There are a lot of different ways to create Gantt charts, I like mermaid.js because it's free and while I've never done it should be able to be automated.
If you're thinking more like visualizing the flow of things of course that's going to vary drastically whether you have explicit locations for things or would have to try and find some automatic way of laying out stations in the flowchart.