That's a myth, fortunately. The Davy Crockett's blast radius was fairly modest, in fact; the weapon was about as small as you can make a nuclear weapon, in terms of payload, and was only like 10-20 tons of TNT equivalent, nowhere near the 150-300 kiloton yields more typical for nuclear warheads today.
The radiation produced by the Crockett was significantly more dangerous than the actual blast, of course, but even that tended to be limited to about 1/4 to 1/3 of a mile in terms of short-term lethality. And while you could set the Crockett to arm close enough to put you in that radius, you'd have to be in pretty desperate straits to take that sort of risk.
Given that even the shorter-ranged variant of the Crockett fired out over a mile... Why wait?