Pretty much
everything within Earth's gravitational field retains its rotational momentum as it leaves the ground, just like the surface it leaves. Do we measure the speed of a tree in the yard or our wash hanging on the line, a bird in flight or a fishing lure cast from shore? Not much to measure there or to see with the naked eye.
That said, there is indeed such a thing as Earth's rotation actually having an effect on objects leaving the surface. It's called the
coriolis effect and it varies by direction and latitude. Germany accounted for this in its calculations when aiming its 'Paris gun' in WW I. Fired from East of the city it's range was seventy-some miles, if not accurate within a whole km or so, but aim accounted for coriolis to move the target
within it's trajectory, thus extending it in practical use. The maximum effective range would be shorter if shooting to the East, as the target would be moving away from the artillery piece.
IMO we may be listening to your opinion(s) more deeply than you are considering ours. The 'proof' we offer is consistent through the ages and universally corroborated. (btw, it lacks race and gender bias, just as Jesus taught) If your 'speculation' is that all governments & cultures have conspired to promote science as though it were a 'religion' since before the New Testament was written IMO it's quite a stretch to imagine that all of them have conspired to promote it.
btw, I'm willing to agree to disagree anytime without disdain, guys.