OK, totally misunderstood your needs.
This video explains why the need for a polynomial counter (starting at 22m15s), but keep in mind that this one had to be a
reversible polynomial counter. A "normal" polynomial counter is usually used as a pseudo-random numbers generator, but it only counts one direction. His random numbers generator had to be reversible:
https://youtu.be/MBT1OK6VAIU?t=1335I never studied the hardware of Atari 2600, so I don't know the details. The only computer I've designed was compatible with Sinclair ZXSpectrum and CP/M, which was already having a memory mapped video RAM, so no need to form on-the-fly graphic sprites from a ROM.
I still don't see why the analog aspect would be important, and not sure why you put "analog" in quotes in the OP, probably it's something else that what I understand by analog simulation of a digital circuit, but I won't insist about it.
To get on topic, QUCS is an open source simulator that can simulate digital circuits. Don't have it right now, I prefer QucsStudio, which is based on QUCS, but distributed as a Windows installer, thought it can run in Linux, too, through WineHQ. If you plan to install any of these, beware there are a few named almost the same, but different products:
https://www.eevblog.com/forum/eda/qucs-qucs-s-and-qucsstudio-simulators-are-not-the-same-thing/QucsStudio can build diagrams like these, if of any help for your presentation:
The attached .sch file is for QucsStudio (after unzipping it).
Apart from the LTspice that is free but not open source, I think
ngspice is a FOSS simulator (based on SPICE), don't have it installed right now either, but it should work for digital circuits, too. I prefer LTspice for its wider support and easier GUI (LTspice can be installed on *nix with WineHQ).
By the title of your presentation, your audience should be comfortable with programming, maybe a few lines of code and a truth table would be good enough to simulate the counters:
https://en.wikipedia.org/wiki/Linear-feedback_shift_register