As I could only remember something vague about extracting things you don't want it was time to turn to the text books. And fortunately for us these points have clearly been pondered for some time as Wolfgang Kunze has an excellent explanation in Technology Brewing and malting:
"The lautering temperature is very important. As the temperature increases the viscosity of the liquid decreases. This means that the lautering would be fastest at 100°C. Although undissolved residual starch is in every case washed out of the spent grains during sparging (continuation of mashing), late saccharification by α-amylase can only occur provided it has not been inactivated by temperatures above 78°C. Lautering at 100°C consequently always results in “blue mashes” (iodine test)."
And then there's a handy which reinforces the important point:
Because α-amylase is destroyed at 80°C it is necessary to keep below this temperature during lautering.
So there you have it, even when you get to sparging you still need some α-amylase activity to prevent starchy wort.