The universe is infinite. FTL just makes it slightly less infinite.
FTL means we actually get to see the grandeur of the universe instead of hypothesizing mathematical models of it. Without FTL, we'll never leave this star system. The grandeur of the universe will be nothing but our imagination, instead of a real thing you can see with your own eyes.
I was explaining to my kids yesterday that it took 33 years for Voyager to leave the solar system, the next closes star is 2,000x further than it’s already travelled. That would require infrastructure to support 100 generations of humans, 99 of which would be indentured by their ancestors to a life stuck inside a space-ark. And it would only require one of those generations to fail for the whole endeavor to fail. And there’s nothing there, it would take another 50 generations to get to what is hypothesized to be a habitable planet.
Human existence doesn’t scale to inter-star system travel.
That's why I choose to believe that FTL will be possible at some point. Otherwise we may as well be the only life in the universe, the rest is dead and pointless. If we can't leave our solar system, our entire universe may as well just be our local system and everything else is just neat wallpaper.
My money is on a quantum theory of gravity unlocking the ability to cheaply warp space.
The Voyager is nowhere near max speed, nor is current biological age the final limit on human life span. If you survive another 500 million years or so, you might meet some aliens then. https://grabbyaliens.com/
If FTL were possible, causality breaks too. Plus if there's other life out there who have discovered FTL, we shouldn't even be here. Don't get your hopes up for more than a tiny portion of the universe ever being explorable.
How big of a lunatic would our great-great-great grandparents have considered us for telling them that soon the several-month trip across the USA will pretty soon take an afternoon, and going to the moon will take 3 days?
Sure, traveling at Voyager’s (impressive, but essentially wagon) speed won’t get it done. But betting against technological advancement has made fools of a vast many.
Generally it's taken for granted that the technological problems with generation ships can be solved with sufficient time and resources. We can probably build a metal box that lasts for a thousand years. We can probably design a sustainable closed ecosystem. We could probably build fusion reactors that run on interstellar hydrogen collected with ramscoops.
But the real problem with generation ships is not technological. Technology can't solve the fundamental social and psychological problems of locking some humans in a box for a hundred generations. That's the most important problem, and the one that's usually waved away with "oh you just can't imagine future technology"
You'd probably need to create a religion for them about some gods/ancients that they are serving in their mission. That seems to be how humans stay focused on long-term social organization across generations.
Mormonism seems well-suited for the religion role you mention since they have the concept of a particular planet being close to the residence of their god: https://en.m.wikipedia.org/wiki/Kolob
FTL means we actually get to see the grandeur of the universe instead of hypothesizing mathematical models of it. Without FTL, we'll never leave this star system. The grandeur of the universe will be nothing but our imagination, instead of a real thing you can see with your own eyes.