Yes it would be nice to know if these things have a reset pin or if removing power is the reset option.
I've never really thought much about this kind of circuit, but I have seen a kind of "disable all" done with a switch.
What I'm seeing in my head is to remove power from all the IC's by having Vss power run through one pole of every double pole switch, which would all be normally closed.
Now, when the desired switch is pressed, it would remove power to all the ICs because it would be an open series circuit. But these switches would have to be momentary, because power would have to be reapplied almost instantly. A learning question for me, is there such a thing as a "momentary pass-through" switch, where a push of the switch would only momentarily break the contact, then reconnect even while pushed? Kind of like a spring that "reloads"? It seems I've seen this before but don't know what its called.
Then, the other pole of the switch, the one connected to the trigger pin of the IC of choice, would also be momentary contact, but it would only engage when fully depressed. This way there is a time delay between the momentary break/reapplication of power to all the ICs, and the trigger voltage going to the IC.
This would be how I would do it if I could identify the proper switch!