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There's no reason that we would expand out at the speed of light in one direction. It's well within the realm of possibility that we can intercept rogue planets or large asteroids to use as long time habitats. Also we can expand in millions of directions at once at sub-light speed. The journey make take a million years, but we'll reach a million places at once.
I didn't say speed of light -- just a significant fraction of it. Even 1% is extremely ambitious from an energy budget perspective. 10% or higher is probably achievable for small outbound probes using laser based acceleration -- but they'll just cruise by systems without any means to stop. For large "settlement" ships or similar, even getting 1% would be colossal amounts of energy (like percentages of the sun's total output). So, yes, you'll need to take the slow road.
Rogue planets come within a few light years of Earth. We could probably have a low speed, multi-generational ship to intercept one in a few hundred years. Once we're on we're hopefully good forever. Likely we'll come close enough to some other interstellar bodies we could populate as we travelled. Exponential growth is bound to take off.
Yeah, if we aren't in a hurry, and we can set up some fusion reactors and such on them and build whole civilizations on these rogue planets in the dark, it would work. Depends on how early and often we set up shop on passing planets, but in theory we could colonize much of the galaxy in a few revolutions around the milky way. So, under a billion years. ;)
I'm guessing if we've reached a level of tech to build a functional generational ship we would be patient.