If they took 100 years to get to the next star system 5-ly away and then 500 years to populate that system enough to start a new colonization, that's just ("just") 168,000 years to be here. Given a 2 billion year head start, that's easy.
Cathedral building in the middle-ages was a multi-generational 80+ year affair and some of the Pyramids took even longer IIRC. So societies were/are definitely capable of working on ~100 year missions :)
Maybe a generation ship needs to have some sort of 'cathedral-in-the-sky' religion connotation in order for us to commit to building it. Sounds like an interesting idea for a sci-fi plot!
But we've only been working on that for a few decades. 2 billion years is a lot longer. If there is a civilization there at all, it's extremely improbable that it hasn't figured out how to do exploration on a scale of thousands of light-years.
I suspect that's unlikely, even if there were advanced life there for 2 billion years.
Why do you think so? "Continuity" doesn't have to be having the same government or the same political structure, or even the same physical form (the article itself talks about uploads and nanomachines). It just means continuity of technological development. It doesn't have to mean steady progress either; our civilization hasn't had that, yet in ten millennia or so we have progressed from basic agriculture to space travel. For this planet, we're talking about two million millennia, or 200,000 times the time we've had. All the variables we've observed in our civilization's history become rounding error on that timescale.