My memory is that 1g of constant acceleration grants sufficient relativity to make it to the edge of the known universe in a current human lifespan.

Now, it's true, there's some slight issues such as radiation, food storage/production, psychological effects, and any random space rocks obliterating your craft, all of which could reasonably turn out to be enough to make it not work. We also don't have a fuel source that can provide 1g of constant acceleration for 80 years for a reasonably sized space ship, though again my memory is that nothing prohibits it from a physics perspective (this is where my knowledge/understanding get prohibitively poor. I'm not sure how the math works if you stick a thousand ion drives to a spaceship that's already in space or if you just need a huge snifter of compressed hydrogen or if you can just use nuclear propulsion but I'm pretty sure that antimatter would do it, if you could bring yourself to waste the money. But maybe we don't have a plausible way to contain it so what do I know).

Maybe I'm remembering wrong, or maybe I glossed over what's currently considered a physics, rather than engineering/economic/materials science problem, but that's what it looked like last I checked.

You don't need new physics for interstellar spaceflight - 16 km/s of dV is enough. you don't even need to go that much faster to slowly spread among the stars. There are a lot of smaller bodies all the way from Sun to Alpha Centauri. As long as you hop between them within reasonable time in a few thousand years you can become a true interstellar civilization, while going at much-slower-than-light velocity (similar to Polynesian colonization of Pacific).

Not with that attitude, we're not!