As other answers note, the history of trying to invent a gender-neutral singular pronoun in English is littered with failures, nonce usages, and usages that become popular in insular communities but don't catch hold in dialect-wide populations. Indeed, while any invented word has hurdles to overcome before it is accepted and widely used, pronouns are an especially difficult class of word to change. Why is that? What would have to happen for an invented gender neutral pronoun to be accepted in English?
How does a pronoun become established in English?
In short: a linguistic need arises and a nearby available tool is adopted to meet it.
First, pronouns have not been invented often in the history of English. The most widely-known example is the importation of they/them/their from Old Norse instead of hie/hem/hier, which is widely documented in texts on Old and Middle English (see e.g. Brown 2009). In this case, the change started with Norse settlement in the north of England, and grew over a few centuries. It helped that the forms weren't that far off from the Old English pronouns, and that they/them/their provided the additional benefit of being clearly distinguished from other pronouns like he, him, and her. So a possible need for disambiguating pronouns was met by a nearby tool - an adjacent language's pronouns.
Newer pronominal forms like yinz and y'all have formed within dialects from the 19th century onward (Crystal 2011). The history of y'all shows how a need (how to distinguish a plural second person after the separate singular form thou disappears from use) is met with adding a modifier (you all), how that modifier was gradually contracted (y'all), and even how that existing form has now sometimes been adopted as a singular pronoun (as Crystal reports: "Howdy y'all" and "Y'all take care now" being delivered to one person).
In contrast, neologistic pronouns usually don't stick around. Pronouns are considered a closed class (ThoughtCo), which means that new words usually don't catch on to the class. That's why I don't have any examples of wildly successful pronouns made ex nihilo.
To adapt this pattern to the current situation:
We need a gender-neutral third person pronoun
The resulting pronoun will most likely be an existing tool adapted to the new situation, since they're joining a closed class of word
What Are the Likely Candidates for Inclusion?
As you note, one word is already being widely adopted for use as a gender-neutral singular pronoun: singular they. Present in English since the 14th century, especially for following singular antecedents like anyone and someone, singular they has also been adopted as a pronominal set by people who desire a gender-ambivalent option. Merriam-Webster describes its growth:
They is taking on a new use, however: as a pronoun of choice for someone who doesn’t identify as either male or female. This is a different use than the traditional singular they, which is used to refer to a person whose gender isn’t known or isn’t important in the context [...] The new use of they is direct, and it is for a person whose gender is known, but who does not identify as male or female. If I were introducing a friend who preferred to use the pronoun they, I would say, “This is my friend, Jay. I met them at work.”
So we have a need for people who don't identify as male or female to have a pronoun suited to them, and a tool that - while not originally nonbinary - did escape the conventional association with a single gender. So people who heard it and didn't initially understand its use would have a better chance of puzzling it out, or remembering its use afterwards. That makes they an effective option.
Variants like themself also show considerable promise, since the form *predates *themselves** but had fallen in popularity in the early modern period (Merriam-Webster). It encounters the problem that themselves, with the plural selves, may be perceived as plural in singular contexts. Having the available historical context, and the well-known word self available, helps the argument for themself, though it has not caught on widely in publications.
Other Pronouns - What Might Stop Them?
Other proposed pronouns include xe and xir, hir, and many others (UWM). They involve changing vowels or consonants to create new and distinct forms. These forms have big hurdles that they need to overcome:
familiarity, or how to make these sound comprehensible and clear upon a first or second usage, compared to the other available options
pronunciation, or how to make the sounds
Xe and hir do okay, at least in writing, because they are relatively close to a recognizable pronoun. With xe, a reader can puzzle out that the h/sh of he or she have been replaced by an x that is neither one. Hir looks kind of halfway between him and her. Neither is quite as familiar as they.
Regarding pronunciation, it may not be clear how to pronounce the x in xe: is it /zi/ or /ɛksˈi/ or /ʃi/? With hir, it may not be clear how to distinguish the vowel from related vowels, like the one in her. Neither are insurmountable with a little teaching, but that in itself poses an issue in a closed system - new pronouns have tended to spread without being formally taught. So it will take some time to see if the proponents of these new pronouns can make the words catch on.
So combined, issues of familiarity and pronunciation may hinder these other pronouns from taking their place in the general population, even if they're familiar to the LGBT community and many of their allies. Even if, in theory, we agree that a totally separate third person nonbinary pronoun would be better, they benefits from being more familiar and easier to say.