The simple answer to your question is 'power'. As in 'because they can', and as in 'power tends to corrupt'.
This makes people, use the rationalization you mention - that we like having them, and so then any excuse will do why it's ok for us and not others.
It's not fair - but people don't really see that, that's the 'power tends to corrupt'.
You bring a persepctive from another side; most Americans don't understand that.
Using an analogy, throughout history of discrimination against blacks, during slavery whites tended to think the institution was benevolent for blacks, even good for them; and during periods of racism, whited did not understand too well the black perspective, why it was wrong. They mainly resented when blacks would rebel against it. One of the common phrases had to do with 'good n*****', meaning one who did not rebel. Same with the phrase 'uppity negro'.
It was an education process - including the historic speech by JFK asking 'would you trade places with a black person' - to help people understand another view.
The nuclear issue is very clear to non-Americans as an unfair policy by the US, and to Americans as the only sensible policy, for them to have nukes and others not to.
There is a grain of truth to the defenders, that the fewer nations that have them the less likely their use is, as long as those few nations don't take advantage of that; and that the US is somewhat more conservative about threatening their use than some other nations would be.
But we Americans tend to not understand the aggressiveness we have had with them, the risks we have taken. We think of the US as a nation that would never use them recklessly.
But our history includes our leading general in a war wanting to use them against China and having to be fired by the President (with the public hugely on the side of the general).
It includes senior US Pentagon officials who have had a low bar for their use, many times actually suggesting it, with one infamous quote summarizing the problem, saying that if there were a nuclear war, and two Americans were left and one Russian, we won. This was less a joke than an actual reflection of some of the thinking of some cold warriors.
We had significant political movements calling for a first strike against the USSR, first 'before the got the bomb' and later 'before they got too strong'.
We put missiles on the border of the USSR, posing a threat that we refused to accept ourselves when they put missiles in Cuba - saying we were willing to go to nuclear war not only over the missiles in Cuba, but even over our refusal to officially agree to remove the Turkey missiles as part of the deal (though we did agree to privately on condition they never say we did.) The Pentagon and CIA and most of the president's advsiors all called for invading Cuba, unaware the Soviet troops there had tactical nukes to respond with.
We had President Nixon running on a 'secret plan' to end the war in Vietnam (after he had sabotaged the LBJ peace talked to improve his chances), where the plan, was for Kissenger to convince the North Vietnamese that Nixon was mentally ubalanced and willing to nuke them if they did not agree to peace (it didn't work).
We've had harmful atmospheric testing, and incidents such as the forced removal of over a thousand native citizens from an area we wanted to explode weapons, where there was a conspiracy to lie that there were no residents in that area, to get around legal protections for their rights.
We've played chicken with the USSR to find holes in their defense systems, in one case resulting in the President being caught in a public lie over Gary Powers - which also resulted in the cancellation of his last chance for arms talks with the Soviets at a planned conference.
We've had a variety of suspected cases of the use of depleted uranium in our artillery.
While there's a case to be made for the use of nukes at the end of WWII, the use was done in a context of very little concern for the civilian casualties, late in a brutal war.
Our firembombing of cities, which Robert McNamara, who was a leader in planning, admitted was a war crime, showed a terrible willingness to kill masses of civilians.
Speaking of McNamara, the military leadership has been so poor at times, that when McNamara became the Secretary of defense, no one outside the Strategic Air Command had ever seen the nuclear war plans, the targets, and he asked to see them and was told no by the Curtis LeMay Air Force, that they were Air Force plans and not his business. He had to get a direct presedential order to see them, and found an arbitrary list of thousands of Societ cities.
Also on McNamara, it includes spending trillions on building tens of thousand of weapons, when McNamara says literally fewer than ten provides a full deterrent strategically.
The history paints a picture far from the sober, conservative nation to claim the right to have nukes that most Americans want to believe.
Rather, the history suggests too much willingness to partake in and condone aggressive, immoral, sometimes misguided military actions, from Vietnam to East Timor to Grenada.
It seems to me that the US doesn't have a good case for claiming the exclusive right to nukes. If a ban can be implemented and ensured by monitoring, that seems the right policy - and reportedly, Obama has committed to that goal at the G20 summit (which is not as new as many Americans may think - the US ha spledged that before, but did not keep its word.)