Is the lay explanation of the equivalence principle wrong?

But doesn't a clock in an accelerating spaceship run at the same rate no matter where in the ship you put it?

Remarkably, the answer is, even in the context of SR, no.

It turns out that acceleration of an extended object is quite subtle.

That is to say, we can't meaningfully speak of the acceleration of an extended object.

Essentially, the 'front' (top?) of the spacecraft accelerates less than the 'back' (bottom?) of the spacecraft if the spacecraft is not to stretch and eventually fail structurally.

Thus, the clocks at the front (top) run faster than the clocks at the back (bottom) as would be the case for clocks at rest at different heights in a gravitational potential.

This is actually well known and best understood in the context of Rindler observers.

Note that Rindler observers with smaller constant x coordinate are accelerating harder to keep up! This may seem surprising because in Newtonian physics, observers who maintain constant relative distance must share the same acceleration. But in relativistic physics, we see that the trailing endpoint of a rod which is accelerated by some external force (parallel to its symmetry axis) must accelerate a bit harder than the leading endpoint, or else it must ultimately break.

Now, this isn't meant to answer your general question but, rather, to address the particular question quoted at the top.

Why does an apple fall from a tree? Why do all objects accelerate towards earth at $9.8$ $m/s^2$? The 'out-of-the-box answer' is that the objects themselves don't move. It's the ground that rushes up! Regardless whether attached to the tree or not, Newton's apple is suspended motionless: it's earth's surface that accelerates up and meets the apple. This simple insight immediately explains why all objects regardless their mass accelerate at the same pace of $9.8$ $m/s^2$.

Right?

Unless you are a 'flat-earther' the above probably comes across like total nonsense. But the point here is that there is absolutely nothing you can do that could convince me to revisit my position, unless you bring into the picture large-scale features such as earth's curvature. That is, you have to introduce non-local effects to force me into acceptance of the concept of gravity. Non-local effects such as the difference in direction between the acceleration of apples falling from trees located away from each other. Such differences in gravitational acceleration represent the true signature of gravity, and are known under the generic label 'tidal effects'.

There is no local measurement you can do on the falling apple that would prove the 'earth rushing up' model wrong. This is, in essence, Einstein's equivalence principle: in any small region of spacetime gravity is equivalent to uniform acceleration.