First, I want to make sure everyone understands I agree these weapons were crazy, and I am glad they were never used. However, historical context is needed to understand the people who designed and deployed these weapons.
1) The dangers of radiation, especially the long term danger, was not understood. The studies that we refer to now were starting when this weapon was designed. Nobody had died from the "Demon Core" or any other test accidents around high dose radiation yet. This was still the period when US service people were exposed to atomic bomb tests at short range to see what happened to them. The idea of worrying about radiation exposure on the battlefield was alien to any of the planners.
2) The planners who requested this weapon had just lived through World War II. They had seen Germany, Japan, China, and the USSR all use masses of troops regardless of casualties to overrun positions that were thought to be impossible to overrun. The political stance was that the USSR had masses of Chinese and Russian troops ready to invade at any moment. Everyone in the west was already primed to fire atomic weapons at the slightest provocation, so moving the decision down to the solder at the front line who had the best view of the problem was just smart in their vision.
3) They already had tactical freefall bombs for aircraft at this level, tactical artillery shells, and short-range missiles. This was just an extension of that idea.
4) Finally, as far as most of these people were concerned, atomic weapons were just another weapon. In the late 1940s-1950s, only a few people saw nukes as anything other than a bigger bang in a smaller package. Seeing the weapons like that, they didn't see any reason not to give the biggest bang they could to the soldiers on the front line.
We can look at this from 70+ years later and think they were crazy, but it all made sense from their perspective. Their mistakes led to the understanding we have now of how and why nuclear weapons are as dangerous as they are and why we shouldn't use them on each other.
