IIRC, it's a legacy of the cold war.
The game theory strategists realised that any use of nuclear weapons would either escalate, or end up in a massive defeat. You can start with a tactical nuke wiping out troops, it will escalate to larger nukes wiping out ships, and eventually you're dropping very large nukes on population centers.
Mutually Assured Destruction was the only solution. (Or peace, but let's be realistic).
The Russians may have gotten away with MAD, without any need for propaganda, but the US is a democracy. To convince the population that MAD was acceptable, they had to create a situation in which any use of nuclear weapons such a horrible crime that it warranted the complete obliteration of the country that dealt the first blow.
In order to make nuclear war (even limited, tactical use) a horrible crime, they made a very big deal out of the effects. (Yes, it's not nice to wipe out battleships and troop formations, but not unprecedented in a war). Which meant they emphasised how bad radiation and fallout can be.
Also, radiation is just naturally a bit spooky, as are many carcinogens (especially ones you don't choose to expose yourself to). The fact that they might kill you 30 years after the event doesn't sit well with a lot of people.