Did the US-China alliance win the Cold War? To what extent did it contribute to the fall of the Soviet Union? I think most of the reasons why the Soviet Empire fell were internal factors rather than external pressures, and the US-China alliance made at best a marginal difference to the outcome.
Indeed, what did the US and its allies actually get out of their anti-Soviet alliance with PRC? Nixon hoped that it would help the US in Vietnam; but obviously it didn't – South Vietnam lost the war and they probably would have lost it just the same even if Nixon's China trip never happened. More broadly, I honestly can't see any military benefits for the US and its allies from the deal. The various potential military hotspots – Vietnam, the Korean peninsula, the South China Sea – would probably be in much the same state if the US-China deal never happened. Obviously some sort of understanding with PRC had to be reached eventually, since pretending ROC was the government of the whole of China wasn't tenable in the long run, but the US could have extracted some major concessions from PRC in exchange for recognition instead of basically conceding to PRC all its major objectives – an alliance against the Soviets was not a huge concession from PRC given that PRC and USSR were already mutually hostile since the Sino-Soviet split of the early 1960s.
Many people say that Nixon's decision was a great strategic move, but I am unconvinced – the US got some short-term assistance against the Soviets which it didn't really need and didn't make much difference anyway; in exchange, the US put itself in a weaker strategic position vis-à-vis China for many decades to come.
And of course, China's economic development since the 1980s has provided many economic benefits to the US and its allies. But, that still could have happened even if there was a more assertive US policy towards PRC, such as a "two Chinas policy" of recognising both PRC and ROC and maintaining diplomatic relations with both. Under this model, it would not be essential that PRC itself abandon the "one China" position, merely that it be willing to accept trade and diplomatic relations with countries that adopted the "two Chinas" position instead. If the US had made that the price for US-PRC trade from the outset, I think Beijing probably would have come around to paying it. In the 1970s, the US was a vastly richer country than PRC (the gap is still there but it has closed a lot in the last 30-40 years), and PRC had far more to gain from US trade than vice versa, so the US arguably had more power to set the ground terms for the relationship – but the US despite having the far stronger hand chose not to play it.