I think part of the problem is that "they" can be unwieldy at times when being used to refer to an individual, and there's no consensus on (or good ideas for) a different gender-neutral pronoun.
A few years ago, Sweden added to "han" (he) and "hom" (she) the gender-neutral "hen". That appears to have taken root (I don't have it to hand, but there was a survey published a little while back which said that most adults now used the word, which is remarkable for a linguistic innovation that's less than a decade old).
But it's easy to see why it would. "Hen" fits nicely with "han" and "hon", as it's just another vowel inside the same two consonants. "They" stands separate from "she" and "he", as do "ze", "xe", etc. because structurally "she" is "he" with an extra letter.*
To make it consistent, really the only thing to do is to add or subtract another letter. "E" has been suggested, and it's not difficult to say or to use in a sentence, but there are plenty of accents that swallow the letter "h" (cockney being famous for it, for example). So that makes it less than ideal.
Adding a letter has two problems - the first being that it usually adds a syllable, which again sets it apart. The second is that adding it to the end changes the character of the word too much to make it really seem like it fits (compare and contrast "she" with "shen"), and adding one to the beginning simply doesn't work (seriously, try to find a letter to add to the beginning that a. is pronounceable and b. doesn't sound silly).
I honestly think that this is a huge factor in which gender-neutral pronouns aren't more widely accepted and used in the English language. Even beyond describing non-binary people it would have uses. But none of them have really taken off outside of certain communities, and I think that a big part of the reason why is that the English language itself makes it difficult to coin one that is easy to say and which feels like a true equivalent to "she" and "he".
I think that "e" is probably the best suggestion. It's easy to say, and you can see the progression "she" > "he" > "e" easily - even if it would be more satisfying if the term for enbies were between those for men/boys and women/girls. It's imperfect, but if there were a sustained push for it in the media the same way there was for "hen" in Sweden, I think it would have some chance of success.
I can understand why "they" was eventually settled on - out of all of the suggestions, it's the one that already existed in the English language. But it's still awkward to say. I try to be careful about the language I use, and have been using "they" for years at this point, but I still find myself often having to put a little extra thought into sentences where I'm using it - especially if I'm using it in the same context that I'm talking about a second party whose gender isn't known.
So I think that "e" has the greatest chance of success if it were to be pushed, although linguistically it's far from idea and I don't think it will be pushed, and I think that "they" will basically become the norm eventually as the younger generations grow up and come to dominate the culture, but not really become truly commonplace beforehand.
*I think that most of the "ze", etc. ones also have the problem of sounding like they're trying too hard to be cool. I changed my name by deed poll some years back and, in choosing a new name one of the things I was very careful about was choosing something that didn't sound like I was trying too hard. If you call yourself something like "Zabrinska Goodfairy" then people are going to go "um, no!". I think that's part of the issue with the "xyr"s of this world.
That's not to say that they can't be used - Greg Egan is a writer who mostly deals in posthumanism, and he uses "ze" all the time. It doesn't seem contrived or odd, but that's a different context to the average bloke down the pub talking with his mates.
