I guess there's a paradox in that we'd only make the progress needed to overtake if we are still launching throughout those 600 years and iteratively improving and getting feedback along the way.
Because the alternative is everyone waiting on one big 600-year government project. Hard to imagine that going well. (And it has to be government, because no private company could raise funds with its potential payback centuries after the investors die. For that matter, I can't see a democratic government selling that to taxpayers for 150 straight election cycles either.)
We can get lots of iterative practice on interplanetary ships, so not much paradox there.
And the research doesn't need to be anywhere near continuous. It's valid to progress though bursts here and there every couple decades.
And a lot of what we want is generic materials science.
Yes, my understanding is that the 600 year figure was arrived at assuming that there is iterative progress in propulsion technology throughout the intervening years. But at the end of the day, it is just some number that some dude on YouTube said one time (although Fraser Cain is in fact not just some dude, he's a reliable space journalist (and you can take that from me, some dude on the Internet))