200+ inputs and outputs: shift registers, multiplexers or other options?

ryanaukes:
I have made some schematics:

Oh, no, you didn't! Those are not schematics. They are "diagrams" but they don't count as schematics. Fritzing does have a schematic view. You can switch to that but you will need to do some tidying to make them presentable.

Do those '4067 breakout boards have bypass caps? If not, you will need to add them. Other than that, your circuits look ok (as far as I can tell until you can give us proper schematics).