It wasn't a "not properly set up" thing, it was the military deliberately pushing back against safeguards they saw as making the US deterrent less credible.
That would work if they made the password full of zeroes public. But did they? I think it's more likely they wanted to increase probability of correct execution. So they prepared for the possibility the correct passwords to launch would not be available when needed, in the simplest way - have a password that is easily remembered.
Those were probably so-called "BCD" switches. You had four data lines because that's what it takes to represent all digits from 0-9, but not all 16 combinations could be dialed up, just 10. The result of putting N switches together in a control panel is a BCD number where N decimal digits are represented with N*4 binary lines.
There are a lot of surplus BCD and other types of thumbwheel switches on eBay, but I'm not sure if they're still being made these days.
Could it be they were using a hexadecimal based system?