When the rocket launches it already moves with earth's rotation at that latitude (and is stationary with respect to the ground). That's in fact the reason why many rocket launch sites are near the equator: Free velocity.

Don't rockets also start with the same horizontal velocity though, since nothing canceled it out when it got off the launch pad?

It would be like jumping, and finding yourself ~250-400 meters away from where you lept by the time you landed.

That said, neat project, and way fun learning experience. Good job.

But the rotational velocity of the rocket is going to be very similar to Earth's at the launch site. Eventually the rotational velocity between ground and the rocket will begin to noticeably differ, but the launch trajectory starting at 408 m/s would look terrifying if it were that way in real life.