Intuitive understanding of scale-space extrema detection

Can someone explain intuitively why local maxima and minima in the scale-space domain make for good keypoints?

I understand using LoG or DoG zero-crossing points to identify spacial variations, i.e. corners. And I understand working at different scales to find scale-invariant features. Both of these are pretty intuitive.

But I fail to see what it is about local extrema that makes them good features for various algorithms like SIFT.

LoG and DoG (an approximation of LoG) masks can serve as blob detectors. A blob can exist in an image at a number of locations $(x,y)$-coordinates and scales (some parameter; $t$). In some situation where scale space is divided into 3 discrete 'slices' and there are only 'small,' 'medium' and 'large' sized blobs, a 'medium' sized blob will have some response to both the 'small' and 'large' sized detectors. However, a detector whose scale is matched to the scale of the blob will have the largest response. For this reason, we want to localize the blob by determining the maximum response in both spatial and scale coordinates.