Why is it a big improvement over 'they'? Singular they has become popular naturally, whereas 'ze' has existed for years without anyone naturally using it(except for trolls on the internet).
If it is good enough for Austin, Byron, and Chesterfield, not to mention Shakespeare, then it is good enough for me.
The catch is that it sounds best when there's no specific antecedent: "Every user of the workshop must be responsible for their own safety."
It sounds odder when it is deliberately used to avoid (mis)gendering a specific individual: "Pat does not discuss their* gender." I think this is the niche that 'ze' is supposed to fill.
I looked at some older examples of singular-they and they almost exclusively refer to an unknown or unspecified person ("Everyone loves their mother"). Using it for a specified person seems to be fairly new: Time Magazine had a piece describing how their (in this sense) was the 2015 word of the year.
I've heard of 'ze' as a gender-neutral pronoun, but I've never really seen it used. How would you substitute 'ze' in that sentence: "Pat does not discuss their gender"?
“Ze” (like “they”) is a subject pronoun, it has been suggested with various object pronouns (like “them”), possessive determiners (like “their”) and possessive pronouns (like “theirs”), including at least these patterns: ze/hir/hir/hirs, ze/mer/zer/zers, ze/zir/zir/zirs; ze/zem/zes/zes.
So, the “their” could change to any of “hir”, “mer”, “zir”, or “zes”, depending on which “ze” pattern was used.
The accord with the verb is still as if it's a plural pronoun. Just like with "you" which technically also used to be plural, doesn't take the old singular second person forms: "you have" not "you hast"
Perhaps more importantly, I don't think singular "they" would be considered proper in formal writing.