If you think this sort of thing is possible you really don’t understand how far it is.
To put it in relative perspective, if we represent the furthest distance humans have ever traveled as 1.3cm then the nearest star is 200 kilometers away.
Of course, the real problem with the analogy is that the fastest you can go 1 cm in this universe is about 8 minutes. So the absolute shortest time it takes you to travel that 2.7km is 4.2 years. That's a few hundred cm per year. But in reality the best we could possibly do is a few tens of cm per year (e.g. c/10), and even that's in the realm of science fiction.
On the bright side, humanity has the Solar system all to itself with wonderful natural barriers to invasion and conquest. So we have time to figure out how to not blow ourselves up and/or stop dismantling our life support system to make Ikea furniture.
1 cm x 60,000 = 600 meters
Which is less than 0.5% of the distance to Alpha Centauri (“200 km” in the analogy, 25 trillion miles in real units)
Edit: a neighboring comment pointed out that on this scale of distance to moon ~= 1 cm, the Alpha Centauri system is more like 1000 km away. So voyager is less than 0.1% of the way.
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.
Moon: 238,900 miles --> 1.3 cm or 0.00 km
(This defines the scale as 0.0000054416073670992 cm / mile.)
Voyager 2: 12,000,000,000 miles --> 65,299.0 cm or 0.65 km
Proxima: 25,000,000,000,000 miles --> 136,040,184.2 cm or 1,360.40 km
The kuiper belt is outside the solar system and then there is the shell of the Oort Cloud that Voyager won't pass through for ages. So it's been decades and we haven't even truly sent anything outside the solar system.
This is all assuming I remembered the terms correctly. Maybe someone who knows this field can comment. Point being getting to another star would take a lot of time :)
Interstellar space is big.
The ISS (1998) travels at 17,400 mph.
There's abundant precedent of accelerating to high speeds in space, where there's no friction with an atmosphere that slows you down and where you can use gravitational fields as sling shots.
0.3c might seem high but if you keep accelerating you will eventually reach that speed.
But there's zero precedent for slowing down once you get somewhere (in 10,000 years)
Which means that a ground station can keep beaming light at them causing them to accelerate.
However the point remains that it’s simply impossible.
My original analogy of 1.3cm versus 200km was wrong according to another commenter in fact it’s more like 1.3cm versus 1,300km - so vastly impossibly far out of the range of anything that could be conceivably be done under any plausible scenario.
Everything is impossible until it's done.
It’s not simply a country we haven’t found yet with a six month sailing journey.
The six month sailing journey sounds waaaaay more ominous.