We would be flung off by the Earth's rotation

"If the Earth spun at 1,000 mph, everything - people, oceans, atmosphere - would be flung off into space. Since we are not, the Earth cannot be spinning."

The outward (centrifugal) acceleration from the rotation at the equator is only about 0.034 m/s² - roughly 0.35% of gravity's 9.8 m/s². That is far too weak to fling anything off; it merely makes you weigh a tiny bit less at the equator than at the poles. The rotation is real and even visible: it bulges the planet, making the equatorial diameter ~43 km larger than the polar one.

a = ω²·R with ω = 2π / 86,164 s (one sidereal day) and R = 6,378 km gives a = 0.034 m/s² = 0.35% of g. Equatorial speed is ~1,670 km/h (1,040 mph). NASA lists the equatorial diameter as 12,756 km; the polar diameter is ~12,714 km - the 43 km difference is the rotational bulge.