A clearer statement is that writing was only invented/discovered in isolation three times that we know of--by the Mayans, the Chinese, and somewhere in the middle east.

Well, Japanese uses syllabaries. The Korean alphabet is a better example of an 'independent' alphabet although it was likely influenced by existing alphabets and there is a theory that it was in particular influenced by Tibetan, which would link it to the "alphabet genealogical tree".

I think you may mean it may have been influenced by the Phags-Pa script [1] (which was influenced by Tibetan, but quite different). Which was invented by a Tibetan and used during the Yuan dynasty.

1 - https://en.wikipedia.org/wiki/%CA%BCPhags-pa_script