We don't need FTL. Our current age limit is becoming more obviously arbitrary every year. Sooner rather than later we'll be able to extend our age limit to several centuries. We could spread out to all the nearby stars in less than one lifetime at even .1c. We could with generation ships anyway, but that's far less desirable. We could populate most of the Galaxy in a million years with no FTL whatsoever, and if FTL never plays out, we probably will. If we meet aliens along the way, then we do.
Your assertion that "our current age limit is becoming more obviously arbitrary every year" is lacking in evidence — it's been somewhere between 120 and 130 years for as long as we can tell, and we have not managed to surpass that even once. That's not arbitrary; it's a scientific fact.
But let's forget that. Let's grant that in the future we'll be able to quadruple the human lifespan to 480 years. For the sake of argument, I'll even grant that manned interstellar flight at 0.1c is achievable. At this rate, it will still take most of our very long lifespans to reach the three nearest systems that are candidates for habitable exoplants. You'll still need a generation ship unless you want a colony of geriatrics. (Also, once you do get there, you're very likely to find that the potentially habitable exoplanets turn out not live up to that potential in much the same way the potentially habitable Venus turned out not to, so you'll spend the rest of your extended lifespan moving on to the next system.)
