Why were the PCs the only people among 10,000 who had a say in the fate of the city?
They weren't the only ones. The ruling council sent the PCs to investigate the alleged crash site of Air Force 1 in the Colorado Rocky Mountains in the hopes that they could find and recover the nuclear football. The PCs had a proven track record of successful missions on behalf of the rulers, and a small force was sent to avoid attracting the attention of the more numerous and better armed cyborg forces from the east patrolling the area.
The PCs were successful in recovering the nuclear football but one of them got cold feet on the way back home. He came to the conclusion that home city was just as bad as genocidal mutants and slaver cyborgs and decided they shouldn't have control of a nuclear device. So in the middle of the night, he crept through camp and tried to destroy it but was noticed by the other PCs and stopped in a most violent manner. The funny thing is that he actually failed to sabotage the NF. The NF was instead rendered non-functional by another PC who thought it was a good idea to add grenades to the struggle. So the PCs returned with a non-functional NF which the city tried and failed to repair. They were able to launch a nuclear device as the invaders approached their city, but the targeting was off and the bomb landed (mostly) harmlessly on a nearby mountain. The combined might of the cyborg and mutant forces overwhelmed the city defenses.