I don't need to stand by the rotation theory. However, the 2.5 degrees that the Earth rotates are about equivalent to the downrange distance.
The first stage is going about 1/5 of the target LEO orbital velocity at separation. While you might well model the trajectory as a parabola over flat ground, given the lack of fuel I would expect that SpaceX puts a lot more care into their trajectory. So far I've failed to attract the attention of the person responsible for Flight Club, the most trusted modeling of SpaceX flights, but I'll message him directly.
Well, Alastair, you should probably not get snotty and ad-hominem, unless you want me to comment on how a one-time sci-fi author and the Unix guy at Dish doesn't really have more authority than the random person one might find in the SpaceX group on Reddit.
It happens there are a few people over there who are rocketry professionals, have the math, and have followed SpaceX long enough. So, sure, their opinion can indeed be trusted.
So far, we have a suggestion from one of the lesser folks there that raising the apogee takes advantage of the Earth's rotation. We'll see if we get the attention of the right people.
Time is an illusion perpetrated by the manufacturers of space.