Until the early 1970s, Taiwan "was" the new China...at least as far as the UN, the US and its allies were concerned. By the 70s (and the Vietnam War) it was clear that the PRC was a viable force with nuclear weapons. Also it had done two things. One is aid the North Vietnamese against the South Vietnamese and American forces. And two is was in a border dispute with the Soviet Union.
While the aid to the NVA hurt and killed a bunch of American soldiers and sailors, the border dispute with the Soviet was "too good" to pass up. It would put a wedge in the Soviet Bloc by keeping the Soviets and Chinese enemies rather than allies. So Nixon went to China. Taiwan was not exactly thrown under the bus, as they still got mountains of aid and military hardware from the Americans to ensure their unofficial independance. but the PRC was now China to the world.
As for the Chinese claim. I don't buy it. In what way were they tricked into giving up those islands (or having them "stolen")? Wasn't that when Formosa was signed over to Japan? So either way the islands would have gone over. That they changed what location administered those islands is an entirely different matter. One that changes who considers it what and where. By the time the Americans came, a generation or two later, they were considered part of the area administered by Okinawa, thus that is were it went by treaty.
Unless someone decides to change that, those islands belong to Japan.
And yes I am quite aware that the Chinese treaty with Japan was to make all their agreements and treaties void as least as far back as 1895. The basic problem was that none of those treaties change the administratio of those islands...that was internal (no idea why they changed it to Okinawa over Formosa. Maybe it had something to do with the shortlived
Republic of Formosa that resisted the Japanese for a few months in 1895 after the
treaty with China was signed).