I'm kinda glad I'm not the only person in the world who remembers that.

The key is implicit in your question: "if we spread to our whole solar system".

If we expand onto the whole solar system to a fair proportion of its total potential, meaning as a species we are a decent fraction of a Kardashev-2 civilization, essentially a loose Dyson Sphere... Yeah, we absolutely can reach other stars.

Using only technology we have today, a set of laser stations powered by the sun (and possibly using it as a lasing medium, if you wanna get fancy) firing a laser at a multi-kilometer laser sail will absolutely accelerate the ship that sail is attached to up to a meaningful fraction of lightspeed.

For arrival, the best design we have is the Orion Drive. A ship with a big Orion Drive to decelerate on arrival, and sped up by a solar-powered laser sail can do 20% of lightspeed. This means 20 years to Proxima, some 40 years to a great many nearby targets. Is that a long trip? Absolutely. But if the ship has good medicine onboard (and it should!) you'd expect most of the original crew to still be alive on arrival.

Importantly, powering one such colonization vessel would take a very laughably tiny fraction of the energy budget available, so it is fully viable to launch such ships by the hundreds, to every star within 10 light-years. That is, i think, 7 star systems.

All of this uses no new science. If scientific progress ends tomorrow, we can still do this! If it doesn't, we'll do better than this.

Correct. That's what light does all the time.