Saying it's not a big deal unless people make it a big deal is not useful. Many people say the same about assuming gender. Language and the way we express ourselves matter and affect the way we think.
There is ambiguity in all natural language and we work around it quite effectively with either assumed knowledge or phrasing which removes the ambiguity (while often adding other ambiguities which aren't plausible so ignored).
Stallman says himself that he can understand his examples, and a couple aren't even ambiguous. He's adding in hypothetical other people to make them ambiguous.
Stallman has highlighted a problem with English and happened to use singular they as an example. He hasn't highlighted a problem with singular they.
Ambiguity –or specificity– has nothing to do with why we talk the way we do. Sometimes, ambiguity is the whole point of a sentence.