>how on Earth (or in space, rather) would we be able to efficiently travel through space?

Efficiently? Or rapidly? They are not always the same thing.

For example, an efficient mode of transport might prove to be quite slow. But if it's efficient enough that means we could send out large numbers of colonists relatively soon in our history -- we would just have to wait a while to see the colonies actually be established.

Whereas a fast travel method might be attractive because of speed but beyond us technologically for another 500 years. Would you rather wait? Or spend those 500 years colonizing existing star systems at the slower more efficient pace?

Anyway ultimately when you think about it, a few centuries isn't very long. Humans spread out across Eurasia at a slower pace than that. It took thousands of years. But we still did it!