Is China's claim on authority at determining who the next "reincarnation" of the Dalai Lama might be any sillier than anyone else's claim of the same?

It is a claim they inherited from the Qing dynasty which legitimizes their territorial claims. The current "reincarnation" process was formalised by the 5th Dalai Lama in agreement with the Manchu Emperor, and its origins are certainly more polotical than theological[0]

Of course Your opinion may differ but I it is no more absurd than the President of the Fifth French Republic receiving the title of the Co-Prince of Andorra alongside a Catholic bishop.[1]

[0]:https://en.wikipedia.org/wiki/Dalai_Lama#5th_Dalai_Lama [1]:https://en.wikipedia.org/wiki/List_of_Co-Princes_of_Andorra#...

This is simple, Dalai Lama by definition, since the beginning of all Dalai Lamas, is appointed by a central government.

Fun fact: The Tibetan monk system is a grand scheme to reduce minority people fertility so ethics population is not a threat to central empire. Search for 黄教 减丁 if you are interested.

So we have two of them: one "appointed" by US, and one by China. For further enlightenment, meditate about the half zero ༳ in Tibetan unicode.

I searched for 黄教 减丁 but all I see are some random Tiebas. Do you have an authoritative source on this?