People hear about dark energy, conceptualize it as a kind of universal repulsion of space itself, and then tend to say that cosmological expansion must be caused by dark energy. Is this correct?
No. Dark energy is not needed in order to explain the expansion of the universe, only to explain why it's accelerating. The successful description of cosmological expansion in terms of big bang models dates back to about 1930, while the effects of dark energy were not observed empirically until around 1998.
Although general relativity does not describe motion in exactly the same way as Newtonian mechanics, the basic insight of Galileo and Newton still applies. Motion tends to continue on its own. In older models of cosmological expansion without dark energy, the expansion of the universe simply continues because of its own momentum, but would have decelerated because of gravity.
Even in newer models that include dark energy, the effect of dark energy is only relevant at fairly late times. Its effects were completely negligible in the early universe.
The cosmological expansion of the universe from the initial big bang is mathematically modelled by the Freedman-Lemaitre-Robertson-Walker equations, which are a solution to the Einstein's field equations (which represents a mathematical description of the gravity, and predicted the space-time curvature). There is not dark energy in that models, which are a correct description of the expansion of the universe until a "more recent epoch" ($z \sim 0.5$). Later in the history of the universe an acceleration was inferred from a detailed study of the light curves of type Ia supernovae.
We still don't know what the dark energy is, but the Einstein's equations plus the cosmological constant term leads to a good approach.