> Why does detection scale as the 4th root of the RCS?

Because the power returned to the antenna scales as the -4th power of the distance to the target, and detection depends on that power being greater than some threshold. If the fourth power (rather than the square) is surprising, think of the reflection as re-transmitting the power that arrived at the target. The incident power on the target goes as RCS/r^2, and the power that makes it back to the original antenna gets another factor of 1/r^2, so the return scales as RCS/r^4.