They were also the "China" entity that was the founding member of the UN. It was only a few years after the UN's founding that the Communist Party won the civil war and formed a new country (the People's Republic of China), yet it took several decades for the UN to officially recognize them as the representative of "China".
The argument to make isn't that they should have a mandate to rule mainland China, nor even that they should be mainland China's representative of the UN. But rather that they should be their own representative at the UN, and have claim to the permanent seat of the Security Council that mainland China currently holds, since it was originally theirs to begin with.
It's a nuclear strategy that'd cause all hell to break loose if that were ever attempted, but is a plausible enough argument that it could likely be forced through if enough parties truly wanted to strip China of their veto power.
In this case, the Republic of China did have a seat, and continues to exist although the landmass they control has shrunk primarily to just the island of Taiwan. They didn't found a new country and call it Taiwan, they're just called that by others for political convenience because calling them by the Republic of China (their original and still official) name unnecessarily provokes the PRC.
The Communist Party officially established a brand new country (the People's Republic of China) once they won the civil war and had control over the mainland. Due to controlling the majority of the landmass originally controlled by the ROC, they were eventually after several decades recognized by the UN as the government body allowed to represent China at the UN (and therefore the permanent seat on the security council held by that position).
But the entity that originally held that seat has never ceased to exist, even if the land they control has shrunk drastically since the UN's original formation. Which is why you could plausibly argue that they have claim to the permanent seat on the security council, rather than the representative of the PRC.
But like I said, that's likely to cause all hell to break loose. As the most direct way for the PRC to respond to that maneuver would be to militaristically assert their claimed sovereignty over the island of Taiwan, and fully dismantle the remnants of the POC that govern it so that it does cease to exist and the above premise no longer holds true. Which would escalate horrifically as other UN member states step in to prevent that. Hence why it's a nuclear option, and not one that is likely to ever happen unless China sufficiently provokes the international community to the point where they see that as an inevitability, and just want to maneuver into a position where they can respond to it from a seemingly unified position without the inconveniences that would present if the PRC still had veto power on the Security Council.
