Without that natural assistance, humans don't have the ability to leave the solar system - we just can't build a rocket powerful enough.
The Jupiter launch window opportunity comes every 13 months so we can do that whenever we want. And we can launch rockets directly out of the solar system if we want; New Horizons had a backup plan of launching direct to Pluto if it had been delayed and missed the Jupiter-Pluto alignment.
That all said, the planetary slingshots aren't nearly enough to cross interstellar distances in a timespan measurable in human generations. We want something like 5% of the speed of light for that, where Voyager is going 0.005%.
It seems like the alignment you refer to inspired the launch of Voyager 1 and 2 because they could visit so many planets at one time, and indeed it did slingshot them at quite some speed. However newer launches have reached similar speeds, and will escape the solar system due to improvements in launch technology.