Einstein's equivalence principle (EEP) states:
"Locally, a free-fall frame in a gravitational field is equivalent to an inertial frame in space in the absence of a gravitational field".
However, the reason justifying this locality is that the gravitational field of any planet converges to the center of mass, and if we consider a freely falling compartment to be wide or high, tidal forces arise. These forces cause the balls released at two different heights (along the $g$-field) in the compartment to recede from each other and cause them to approach each other if released perpendicularly to the field.
On the other hand, it is mathematically allowed to assume a uniform gravitational field, which excludes the said deficiencies as to the tidal forces. That is, if a compartment of whatever width or height freely falls inside a uniform gravitational field, it is anticipated the entire compartment to be inertial, and the observer located wherever within the compartment by no means detects whether he, as well as the compartment, is floating in an interstellar space or freely falls inside a uniform gravitational field.
However, what is upsetting here is the difference in the gravitational potential of the clocks located on the floor and ceiling of a high compartment. Indeed, even in case of a uniform $g$-field, the observer located on the floor of the freely falling compartment detects that the clock on the ceiling runs faster because it is in a less negative $g$-potential compared to his own clock. Therefore, the observer can distinguish if he is floating in an interstellar space or being fallen freely in a $g$-field.
My question is why many textbooks refrain from explaining the main reason for the locality of EEP? It seems that it is not about tidal forces but rather the $g$-potentials.