The thing I hate about that phrase is it assumes technological progression magically stops for a BILLION years (then the sun becomes too bright and boils our water). If we go extinct it's not because the sun exploded but something else. If we survive the next billion years, we are going to be in fucking space.
You dont have to go faster than light to get to another star. It's not easy but you couldn't possibly imagine a billion years of technological development.
You mean to say millions of years. The tech needed to accelerate enough to make the trip in hundreds of years is also far outside the realm of physics as we know it
The orion drive is basically an engineering challenge. Very little new tech required. It could reach nearby stars in hundreds of years with no new physics.
The problem isn't travel time, it's speed. Space itself is continuing to expand, at a faster and faster rate. It's expanding so fast that we'll actually never reach far out with conventional means. We'd just endlessly drift through the darkness because we wouldn't be able to go fast enough to reach anything. A generation ship would simply not be able to get anywhere, ever.
I mean, maybe we can make it into another very close system, but what are the chances that there's anything even close to being habitable?
On a local level the expansion of space cannot overcome gravitational attraction to a certain scale, roughly around the size of our local galaxy cluster. We'll always be able to reach anything in our local galaxy cluster without FTL travel.
Your sense of scale is off. The amount of stuff held together by local group gravity is more than we can ever get bored with.
The amount things that we can reach without FTL is so large that humans could explore and colonize for billions of years without ever feeling like we are running out of space.