The earth's axis of rotation has to pass through its centre of mass. There's nowhere else that it could pass through.
No matter where I attach the thread, exactly half the mug will be on one side and half on the other. The centre of gravity is directly below the string in each case. If I could be bothered to set up a more orthographic view and take a load more photos from a fixed position, it would become pretty clear where exactly the centre of mass is. But that isn't especially interesting.
The fact that my mug isn't a perfect sphere doesn't stop it having a centre of mass, nor does it stop it rotating quite happily about any axis which passes through that centre of mass. Now, if you tried to rotate it about an axis which didn't pass through the centre, you'd get a wobble. Like a misaligned wheel. But the earth isn't skewered on a cosmic axle, it's freely rotating in space. So it can only rotate about its centre of mass.
Your explanation is almost correct in every sense, but there are a few details that have to be discussed now, before an FE'er with no understanding of Physics starts to make a fuss.
First, your plomb does not
exactly point to the center of gravity of Earth. The difference is minuscule, but measurable with the best equipment available. For example, if you measure the angle between the line of the plumb and a true vertical (using stars as your point of reference) while you are on the side of a heavy, rocky mountain, you will find an angle different from zero. Your plumb will show the approximate location of the CoG but not the exact place. In fact, for some time we had more than one estimate of the height of Mount Everest because the triangulation from different places gave different results.
Also, when you spin a non-homogeneous object from its center of gravity there may be a wobble. Take, for example, a car wheel with a dented rim. You measure the center of gravity and it is pretty much in the center of the rim. But you spin the wheel at some 100 km/h and the whole car shakes. By putting weights on the rim the dynamic balance is restored even though the static balance is slightly affected.
All of the above has affected Earth's spinning and makes it a very complex movement if we look at tens of thousands of years at a time. But because most of our planet is liquid magma, these effects are small. On the other hand, the slight imbalance of the Moon, and its lack of liquid magma, have contributed to its synchronization with Earth, making one side of the Moon face Earth all the time.
Now, back to the thread, nothing of the above affects the simple, straightforward experiment of the Foucault's Pendulum. The effect of these minuscule differences is orders of magnitude less than the typical one degree or so that Foucault's Pendulum moves.