One, they worried about whether we'd risk New York City to defend West Berlin. That is, would we really use nuclear weapons against the Soviets if they were defeating us in conventional war when the price might be a crater where New York City is? If we weren't willing to do that, the Soviets would be more likely to invade NATO.
Two, they worried we would use our nuclear weapons and turn central Europe into a nuclear battlefield under some tacit agreement with the Soviets not to escalate to attacks on each other's national territory.
France and Britain had independent nuclear forces that could be used that would trigger America's use of nuclear weapons should Moscow then use them. The logic might be insane but it fit the need for deterrence. If we wouldn't nuke Soviet territory in the face of Soviet conventional victory, France or Britain might. More uncertaintly was added to the question of whether the USSR could launch a conventional invasion without triggering nuclear war.
We had even more nuclear linkage than I realized (or perhaps this is something I once knew--or assumed--but forgot). Indeed, we still have 200 thermonuclear bombs in Europe that, in theory, the Belgians (!), Dutch, Italians, and Germans could use:
Under a NATO agreement struck during the Cold War, the bombs, which are technically owned by the U.S., can be transferred to the control of a host nation's air force in times of conflict. Twenty years after the fall of the Berlin Wall, Dutch, Belgian, Italian and German pilots remain ready to engage in nuclear war.
Disarmament types are in a lather over this.
The need to link our nuclear arsenal to the defense of Europe doesn't seem terribly urgent, I admit. However, while they are unlikely to be needed against the Soviets, it might be good for the Iranians to know that Europeans might get access to enough bombs to flatten them should Iran want to threaten Europe with a nuclear sea of fire.