A "dreadnought" has as its definition a rather singular quality: A capital ship with uniform heavy armament.
In this sense, USS South Carolina qualified. No question.
But as you point out, the impact of HMS Dreadnought was what it was because of more than just its uniform 12" armament. It also was the first to incorporate steam turbines, giving it an unheard of (for capital ships) 21 knot speed, and a generally much more sustainable power plant. It was also one of the first to use electrical range finders. It had a new watertight bulkhead scheme. And yes, at the time of commissioning it was, I believe, the heaviest battleship afloat.
But one other splash it made was a literal one: the record time in which it was laid down and commissioned - basically under 14 months. USS South Carolina, on the other hand, took three years and three months, only commissioning in March 1910. That speed was a function both of unmatched British shipbuilding capability as well as Jackie Fisher's prioritizing Dreadnought's construction over all other ships.
So: even though a number of powers and naval experts were considering all-big gun ships at the same time - Cuniberti's and Poundstone's articles both appeared in 1903, Fisher had already been thinking along these lines, and South Carolina was being designed at the same time that Fisher's Committee on Designs was designing the Dreadnought - the Royal Navy beat everyone by a few years. Which means just getting the US Navy the funding a year earlier won't suffice. There must be a much greater sense of urgency to the US Navy, and it's not clear to me how that happens short of a war scare. As it is , the moment Fisher is installed as First Lord (Oct. 1904), he will work like the devil to beat any other naval power to the punch. You'd almost have to have South Carolina slipways before he takes office.
Always nice to have the fastest shipyards.