A datapath holds a value in a register, adds a step to it each clock, and displays the result - drawing on gates, adders, registers and timing at once. Why is understanding each piece separately insufficient to design the system?

B. Because the pieces constrain each other: delay, width and settling time.

The pieces are correct individually and coupled in their requirements: widening the adder for a larger range lengthens the carry chain, which lengthens the critical path, which lowers the clock, so a decision about range has become a decision about speed. Which is why hardware design proceeds by iteration, since the critical path is a property of the whole rather than of any part.