Structured programming allows only sequence, selection and iteration, and it was proved that these three suffice for any computable function. Why did anyone need to prove that, and what was being given up?

D. Because the alternative was goto, and removing it costs no expressive power.

Böhm–Jacopini mattered because it removed the obvious objection: programmers could reasonably ask whether dropping goto made some programs unwriteable, and the theorem answered no. What is given up is arbitrary jumps, and that is the point, control enters a block at the top and leaves at the bottom, so a block can be understood alone. The argument was about human reasoning, not machine behaviour.