All well-made points, Sammitch.

In the case of the U.S., though, ironically we should be among the most secure nations on earth, because our having the largest nuclear arsenal should mean that no one would want to mess with us.

But (again, ironically) the reality is, we among nuclear-armed nations have the greatest conscience and reluctance that would prevent us from actually using them in either a pre-emptive defensive or retaliatory way.
On many levels beyond nuclear weapons, the U.S. is seen as a paper tiger that is afraid to use its power and influence. Where despite the U.S.'s enormous and unmatched power, even piddly nations like Iran, North Korea, Venezuela, Mexico and Cuba seem eager to take us on, knowing they can defy us with virtually no risk of consequences.

Similarly, Osama Bin Ladin became a household-name hero throughout the entire muslim world for defying the U.S. with his early (pre-9/11) Al Qaida attacks, and even when we bombed him, he survived and grew as an icon of muslim defiance against the U.S.
Showing that even U.S. military action, when it doesn't have follow-through to get the job done, is not a deterrant. Limited action actually made the threat grow.

Our limited threat (either conventional military, or nuclear) likewise just emboldens their defiance and will to do us harm.

I think you hit it especially right that Iran and North Korea want nukes for (1) a bargaining chip, and (2) a hedge against invasion.

But like China, India and Pakistan, these nations also seek nukes as (3) an elevation of their national prestige by becoming a nuclear power.
And in the case of Iran, (4) advancing the cause of Islam by actually using nuclear weapons, to accelerate their Islamic visions of Armageddon.
Unlike the "mutually assured destruction" that kept NATO and the Soviet Union in check, Iran has the zeal to actually use their weapons.
North Korea knows their government would topple immediately if they ever used nukes, so I see the greatest threat from North Korean nuclear capability to be their potential spread of nuclear technology to Islamic nations and Islamic terror groups.