I don’t believe it’s as much a matter of won’t as can’t.
Studies were done during the Bush era that looked at the specific issue of salted snacks and soft drinks. They found that food stamp customers purchased less of these (both in the absolute and as a proportion of their spending) than cash customers did. This was one reason why the programs did not bother to ban these items, as requiring retailers to identify and exclude them would raise the cost of the programs.
How hard can it be when it is all digital? A banned item gets scanned and is kicked to a “cash” list or is excluded. That doesn’t do anything to the cost of the program.
Its funny: Costco can tell me how many of something I bought in a year but we can’t keep up with food stamps? We can definitely know where the money is going if there is the will to know.