[leni536]

This is a very good intuitive explanation.

For a quantitative explanation one can make use of the fact that angular velocities or infinitesimal rotations compose like vectors. Earth has an angular velocity that is parallel to its radius to the North pole. When you look at the movent of the surface of the Earth around an arbitrary point, then you can decompose this angular momentum to parallel-to-radius and parallel-to-surface components. The former describes how fast Earth's surface "twists", the latter describes how fast it tilts under your feet. The former is the relevant component for describing the pendulum.

Interestingly this angular-momentum approach has a deeper connection to your car analogy. It can be used to solve the following problem: The Little Prince buys a little car that can turn around on radius r on Earths surface. He brings this car to his home planet, which has radius R. On what radius can this same car turn around on this planet?