> There were lots of administrations who could have said to other countries ,,let's get rid of the nukes together'' while USA was the only string power.

There was only one administration with that opportunity, really; Truman.

Every other administration has had a nuclear armed Russia in play.

Attempts to do what you describe were still quite common, starting as early as the 1950s. https://en.wikipedia.org/wiki/Nuclear_arms_race#Treaties