No.
Salt is not fine enough to cut cockroaches.
Cockroaches have exoskeletons that protect them from sharp objects.
And salt is not sharp enough to cut through them.
Whether ground, flake, or crystal, no salt is sharp enough to cut cockroaches.
The idea that salt can cut cockroaches is likely a misunderstanding between a standard product used to kill cockroaches – Diatomaceous Earth.
DE is sharp enough to cut cockroaches, but this is not the primary way to kill them.
First, DE absorbs the oily and waxy outer cuticle layer found on a cockroaches exoskeleton.
Once the outer layer is destroyed, cockroaches lose water and die from dehydration.
Although to humans DE is just like any other powder, to insect, it has microscopic, razor-sharp edges.