Milord /u/ThePeasantKingM ,
The idea of seating the PRC on the UN Security Council, instead of the ROC, went as far back as the 1950s in the early stages of North Korea's invasion into what became the Korean War. Over time, most countries started to accept the idea that the PRC should be a member of the UN and its Security Council, with the notable exception of the US supporting the ROC's stance against this. Political changes in the late 1960s-early 1970s, both in the PRC and in the US, led to significant re-alignment in 1972 whereby the US removed its opposition against PRC membership in the UN Security Council.
In terms of process, things came to a head in 1971 when Somalia -- then temporary member of the Security Council -- brought about an objection of the ROC's status as China's representation to the Security Council. This resulted in both the US and the ROC asking that this question NOT be dealth with in the Security Council. Therefore, the issue ended up being debated in the General Assembly. Thus, the ROC was not able to exercise its veto power.
Now in more detail.
In the mid 1950s, the USSR was boycotting the UN Security Council, due to the refusal of other members of the Council to seat the PRC instead of the ROC. Ironically, this boycott meant the USSR was not able to cast its opposition to UN resolutions condemning North Korea's invasion. There was some real fear that the PRC would use the resulting chaos and panic to invade Taiwan, thus Truman ordered the US Seventh Fleet to patrol the Taiwan Strait.
At this stage of the Korean War, the PRC maintained an ambivalent stance. It blamed South Korea for being aggressive towards North Korea yet it downplayed the magnitude of the war itself. It remained neutral and made little of North Korea's early gains. Its press expressed concern that US military movements into South Korea was a regional threat, yet it moved some of its troops in Fujian -- troops that had been there with the intention to attack across the strait into Taiwan -- to the northern regions both in recognition that they could not invade Taiwan with US presence nearby and also to downplay their threat to Taiwan.
In early August 1950, when the situation looked grim in the last stand of South Korea and US troops in the Pusan Perimeter, the UN facilitated talks with the PRC to raise the prospect of PRC membership on the Security Council if the PRC will help mediate the conflict. So, there was already some acceptance that the PRC should have the seat on the Security Council, in particular due to the declining power of the ROC itself.
However, at the same time the supreme commander of Allied forces commander of UN forces in Korea Gen. Douglas MacArthur
held his own meetings with Chiang Kai-shek in which he declared Taiwan to be an integral part of US defense and security chain in East Asia.
More strongly, Chiang offered the deployment of ROC troops in Korea, but this was turned down.
Regardless, all this turned the UN negotiations with the PRC hopeless, as the Korean War and the issue of PRC-ROC was turning into one political conflict.
By late August 1950, the US Congress approved emergency measures and funds, and the US/South Korea forces started to beat back the North Koreans. This turned the PRC rhetoric from ambivalence to outright aggression. The eventual PRC entry in the Korean War turned its relationship with the US into enmity until the late 1960s, during which time the two had limited diplomatic contact. In addition, internal challenges in the PRC due to the Cultural Revolution made improving relationships difficult.
It is during this time that the US and its allies advocated for UN Resolution 1668 at the General Assembly in 1961, which states that proposals for change of recognition either to the PRC from the ROC, or the representation of all of China at the UN, require super-majority (2/3).
But toward the 1970s, the PRC started to be concerned with its own relationship with the USSR. Criticisms between Mao and Krushchev, in addition to unresolved border issues that spilled into popular arguments, led to deployment of troops that culminated in armed conflict. Finally, rejection of USSR expertise and internal economic crises meant the PRC was becoming dependent on its oilfield production, which was in serious decline. China has always been an oil importer, but discovery of new oilfields in the 1960s fueled speculation that it may become a major exporter if expertise and investments were made available. In the worst times, the oilfield industry was the only growth industry in the PRC.
Thus, PRC leadership started to change their rhetoric in the 1970s, and started to broadcast their willingness to communicate directly with the US. The US, of course, was interested in taking advantage of the Sino-Soviet split and wanted to gain access to China. By this point, most other countries had moved to support the PRC's wish to be a member of the UN and have a seat on the Security Council, with the US being the notable major power opposing this. Thus, US support of a change in policy, under Nixon, removed nearly the last hurdle.
At this point, discussion of what was to happen to the ROC came about. Both the ROC and the PRC was strongly insisting on the One China vision, thus a compromise became impossible. Thus, the UN General Assembly Resolution 2758 effectively giving the seat to the PRC specified strongly that it was a “restoration of the lawful rights” to the PRC. This suggests that the PRC should have been given the seat in 1949, and thus avoided the need to obtain the supermajority required by UN General Assembly Resolution 1668 if a change of recognition was in question.
Of course, there was an effort to prevent Resolution 2758 from being voted without requiring a supermajority, however this was defeated narrowly. In response to this defeat, the ROC declared that this failure amounted to a flagrant violation of the UN Charter, and thus the ROC declines further participation in the General Assembly proceedings. Thus, the ROC lost its representation in the UN as it withdrew.
Spence, The Search for Modern China, 1990.
Meidan, The Structure of China's oil industry: past trends and future prospects, OIES Paper WPM 66, Oxford Energy Institute, May 2016.