Right there, in the first paragraph, you find the first and worst mistake of all: they forgot the word "locally".
This might seem to a Relativity illiterate as useless nitpicking, but it is the difference between real science and cheap philosophy.
If you get into a closed box, with no possibility of seeing or detecting anything outside the box, the principle holds. But as soon as you have a scenario where you can see the effect of gravitation in different (not local) places the whole equivalence principle ceases to hold. For example, when the Earth, Moon and Sun interact with a very large ocean, the ocean does not fit into the "locally" part of the principle, and unsurprisingly does not act as if it was just accelerated in one direction.
Similarly, in the Cavendish experiment, the interaction of the gravitational forces of Earth and each of the weights is clearly distinguishable from a single acceleration in one direction.
Also, if you are measuring the location of Earth, Sun and any planet, you will see that the gravitational pull is clearly distinguishable from a single acceleration in one direction.
And this is just a very short list of cases where there is no locality and gravitational pull is clearly distinguishable from acceleration.