As far as I've read the literature from the 60s and 70s, tactical nukes were eventually eliminated in order to assuage western Europe's concerns that large portions of their countries would be turned into irradiated wastelands for decades / centuries if war erupted between the US and USSR.
It was also the product of perceived overmatch on both sides -- the Soviets believed they had superior mass of armored formations (and they did), while the US and allies believed they had technological supremacy (and they did). Ergo, neither needed tactical nukes.
It didn't hurt that it helped both in the eyes of the then vehemently anti-nuclear European movements.
Offensive bio and chemical weapon limitation is a more nuanced decision.
In both cases, their primary use was either local mass lethality or terrain denial, neither of which were important in the then-gelling American doctrines of maneuver.
The sole use case they seemed viable for was industry denial (e.g. contaminate a high capital cost industrial center), a task at which strategic sized nuclear weapons were equally adept (and more easily stored). So, if you had to have strategic nuclear weapons for deterrence, and they were capable of the same task, why have fiddly bio and chemical weapons?
But in both cases there was also a constant radiant pressure of scientists and the public campaigning against them, and being unwilling to work on or tolerate them.
Absent that, who knows how history would have turned out? Normalization is a powerful opinion shifter.