Whether a nation or people is sovereign or not is a statement of reality, and part of sovereignty is whether other sovereign countries are able to vouch for your sovereignty. Just formally acknowledging your sovereignty like having an embassy is the bare minimum, but there's more like defense treaties, economic deals, and joining organizations. At the end of the day, there will be other countries and entities that will seek to challenge and destroy a country's sovereignty and unless you have a fleet of Gundams, you'll need other countries to rush to your defense when it's challenged. Nobody can do it alone.

If you're talking about cases like the ROC being in the UN instead of the PRC even though the PRC is de facto far larger than the ROC, remember that the UN isn't an immutable organization. There's nothing stopping the ROC from denouncing the UN as a sham organization after getting kicked out and starting their own rival organization called the League of United Nations or something and getting other countries to cosign to this new organization. There's nothing stopping the ROC from campaigning their allies (ie the West and various Western vassals) to leave the UN and join the LUN. But for obvious reasons, a LUN would never happen because the ROC doesn't have many allies. Most countries, including its so-called allies, see Taiwan as a US unsinkable aircraft carrier that'll inevitably be reunited with the Mainland, whether peacefully or by force, or be completely destroyed in the process of a US-China war.

Agreed.

Wait, they took their ball and went home and you're defending that as a show of legitimacy?

Taiwan is currently a Chinese settler colonial regime that functions as an American military base and microchip factory. If they want sovereignty, they should give the land back to the natives they stole it from and return the billions of dollars worth of gold and artifacts they looted from the Qing coffers after they (catastrophically) lost the easiest civil war in history.