Desmond nailed it.
Smoothing in the Apple infrastructure seems to work only with neighboring pixels. therefore, if you reduce the image by 4x, a pixel of 0.0 will be averaged over a pixel of 2.0, i.e. pixel 1.0 is completely discarded. therefore, if you have sharp edges, this will not crack it.
The solution is to resize the image by 50% until it becomes <2x the desired size.
Here's what happens if I gradually reduce the original image to 600 x 600, halving it each time and showing a result of 256 x 256 CALayer:

The third image actually shrank below the target size. It is strange that further reduction (which automatically gets a size of 75 β 256 for display) actually looks the best.
So, in this case, the solution was to reduce it until it was actually smaller than the required size, and then let Apple's graphics structure expand it as needed.
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