There were various reasons why our grandparents sought to limit the availability of nuclear weapons in general and in particular among even our allies. I can think of three.
First, during the Cold War there was a notion of a solid-front, anti-communist “West” led by the U.S. and Europe, and ultimately defensible by the nuclear capabilities of the U.S. and to a lesser extent France and Great Britain. The small number of Western powers with a nuclear deterrent to the U.S.S.R. had the effect of grouping by needs other allied, but non-nuclear nations (e.g., Germany, the smaller European states, Japan, South Korea, Taiwan, Australia, etc.) around a common cause and defense. Although France often freelanced, ultimately the Soviet Union accepted that French deterrent forces were firmly in the U.S. camp as part of a collective NATO alliance. Such solidarity transcended the post-Cold War age and is still valuable against new radical Islamists and their sponsors as well. The restriction of nuclear weapons helped to solidify and focus Western agendas.
Second, there were lots of anomalies in the nuclear club. A strong Germany did not go nuclear; yet, its historic rivals and often weaker states, France and Britain, did. Like it or not, the causation of three European wars was eliminated, in the sense that non-nuclear Germany did not seek to transform its ascendant economic dynamism into military power or territorial acquisition. The same was true of Japan. An accepted consequence was that the neighbors of the old Axis powers, for the first time in decades were relieved of their ancient worries. Age old-tensions between Japan and South Korea (or for that matter Taiwan as well), or Germany and France largely grew dormant. Today’s EU tensions between a creditor Germany and its European borrowers, or Japan's new muscularity that worries South Korea would only be magnified with the expansion of the nuclear club.
Finally, in a narrower sense, U.S. security was enhanced by limiting the number of nations, both friends and foes, with access to nuclear weapons. By assuring allies that they reside inside the American nuclear umbrella, we have reduced the chances that we will have our version of a renegade North Korea that is both used by, but also at times ignores, China. Constitutional government is a frail enterprise and we never know which stable ally today may prove unstable tomorrow; such volatility is manageable if they remain not nuclear.
The key to such geostrategic stability is the constant reassurance from the U.S. that our allies, which certainly have the technological savvy to make nuclear weapons quickly, do not need them, given that their cities are as sacrosanct as ours, and their defense needs dovetail with our own. Any fissures in that once granite-like commitment, real or imagined, will unwind the entire postwar system. The Western version of the dissolution that we are now witnessing in the Middle East would be an escalation of bickering among our allies to the point that some chose their own strategic defense—mostly in worries that the old U.S. guarantees were no longer iron-clad.
Let us pray that present flirtations with “lead from behind” and doubts about American exceptionalism remain rhetorical and do not convince friends abroad that we freeing our allies to deal on their own with regional hegemonies like Iran, Russia, or China.
The result will be chaos unlike we have seen since 1945.