I'm entirely sympathetic with his preference for precision when at all possible. One of my great pet peeves in modern usage of language is the lack of precision of usage, which I think leads to a lack of precision of thought and communications that creates bad feeling and wastes time unnecessarily. When people are lazy about using words correctly they can also become more generic over time in general, which in turn robs other people of the ability to get their thoughts across as well down the road.
However while I think his examples of where "they" can lead to ambiguity are fair, his argument that "we have a clean, clear and natural solution" is not supported by his own examples. Take the first one for they:
>"When my child was removed and placed with Dad, they internalized it and took it like they did something wrong."
Let's try his "natural" answer:
>"When my child was removed and placed with Dad, person internalized it and took it like pers did something wrong."
What. That's awful. Certainly no better then "ze internalized it" or any other entirely made up word. "Per" and "pers" and "perse" are not "equivalent and interchangeable" with person in normal English at all, and in fact "per" is already in usage as a preposition. He does this whole setup but then his solution falls hilariously flat, and in fact illustrates precisely why people use "they".
If people just want to bite the bullet and try to push a universal gender neutral singular pronoun like "ze" or "zhe" or whatever for English I don't have any problem with that. I think it also might be practical to achieve, it's just "the singular of they" so plural doesn't change and nobody ever needs to think about it again. It has the advantage of laziness, everyone can just start using it everywhere and never have to think about it again. I don't think it can really be pretended that anything existing can just slot in there though, certainly not what RMS came up with.