Is there any intuition for this?

That depends on your intuition. A person who has studied special relativity knows that accleration is velocity dependant. I.e. let a' and v' be the acceleration and

velocity of a particle in S' and let a and v be the acceleration and

velocity of a particle in S. Then a is a function of both a' and v'. In GR there is a bit of a difference in that the gravitational acceleration (which is an inertial acceleration) is a function of v.

I know if you were trying to accelerate a craft from rest using a rocket, you get diminishing returns at higher speeds (with respect to dv/dt WRT the rest frame) due to approaching the speed of light. Is it at all similar to that?

No.

Something occured to me after I started this thread. The vlue of the 3-velocity can have different values. If we use the coordinate value r where r is the r in the Schwarzschild metric then we'll get a different value that one would get if we used the distance traveled. So it's much easier to show that dr

^{2}r/dt

^{2} is a function of dr/dt. I'll try to work it out in terms of the physical displacement dL (aka "shell" distance) rather than dr which is a coordinate displacement. One also has to distinguish which time one uses. In the first version of the text the authors used local time (or "shell" time).

The text asks the reader to derive an expression for the gravitational acceleration using an initial "shell" velocity, i.e. dr_shell/dt_shell, of zero. This problem is the same with only one fact changed, i.e. the initial "shell" velocity is not zero.