Why would the electrical system cut out completely when starting a 1989 Chevy Celebrity?

It could be as simple as a corroded battery terminal. Remove both battery terminals and clean them then see if it makes any difference. BTW, when a battery starts to corrode terminals it usually means that it's starting to vent some battery acid during the charge cycle. That's a sign that the battery is getting ready to go bad. It's probably still ok for now, but be ready for it to go out on you.