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).
Perhaps more importantly, I don't think singular "they" would be considered proper in formal writing.
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 suspect it depends on dialect. "their" just sounds more informal to me, "his"/"her" sounds slightly more formal.
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.
So, the “their” could change to any of “hir”, “mer”, “zir”, or “zes”, depending on which “ze” pattern was used.
> The Oxford English Dictionary traces singular they back to 1375, where it appears in the medieval romance William and the Werewolf.
from https://public.oed.com/blog/a-brief-history-of-singular-they...
There's not a man I meet but doth salute me/ As if I were their well-acquainted friend.
It does sound wrong for some uses of he/she.
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"
“I’m worried about my kid at summer camp”
“Don’t worry, I’m sure they are having a great time”
Sounds very natural to my west coast ears.