As our species is more than likely to self-exterminate within the next 200 years, it is most probably redundant to think about our small-brained ape species going further than the reaches of our own solar system. The real problems aren’t technological but biological. We’re highly adapted for our own planet both in terms of physiology and psychology. Our modern world already creates huge psychological problems by placing us in non-natural environments like cities. Imagine what would happen if we were stuck in a tin can our entire lives.
And we can always count on fundamental human stupidity to cause problems. Here’s why we will never colonize even a tiny fraction of the galaxy, even if ER bridges didn’t expand faster than you can traverse them, even if space-warping didn’t create irreconcilable problems with quantum gravity, etc.