It's not the star, it is the planet.
Even if we don't destroy it, there's a dozen natural disasters that can end the species if we don't expand.
Beyond that, the nature of human cultures is expand or stagnate and collapse. Add in the inherently tribal nature of the species and conflicts are inevitable.
We need room to separate the tribes. A pressure valve so we can focus outwards and not on each other.
The moon, Mars, Ganymede, Titan, and Callisto can serve as temporary targets but only if we maintain a reasonable level of technology. Less than star travel but better than sticking to the mudball.
Problem is, that still leaves a half dozen ways to go extinct. Pandemics , supervolcanos, supernovas, solar instability, large interstellar intruder, say a brown dwarf altering orbits. Or, yes, wars. (But not aliens, thankfully. They make for good SF but the odds are we are alone.)
Humans fight.
Humans kill.
Especially when a lot of them are too close together.
Cities are near the top of the list of worse human inventions.
Best to keep our distance from each other.
Trick is find a few dozen worlds to spread out to until our genetics move past our inner troglodytes. Tech moves faster than evolution.