When you go outside to look at or photograph the moon, you don't face East and then crane your neck straight up until you see the moon. You face whatever direction to see the moon most easily.
When the moon first rises you will be facing East, and when it is about to set, you'll be facing West. Half way between, in the northern hemisphere you'll be facing South.
One quarter of the way through the moon's path, you'll be facing Southeast. You are turning to face the moon, and the Earth's rotation is tilting you.
In the Karen B video, when attempting to claim that equatorial mounts don't work, she shows an animation of why the sun should not rotate, but the animation shows incorrectly the camera perspective is locked to the North Pole, it doesn't rotate with the Earth as it would in real life. I'll try to make you an image to look at.