Mostly, because they already have singular they. It's been in the language since the 14th century. Prior to that, there was generic he, which continued to also be used until the 20th century and is still found. There are also the inclusive doubles "he/she" and "s/he" ("inclusive" though not as inclusive as we might want, on which more below).
The earliest reasons for wanting an alternative to both singular they and generic he that was an explicitly third-person, singular, gender-neutral pronoun came not so much out of sexual politics (whether to ignore gender where it isn't known or relevant, or to describe non-binary people) but the 18th century dislike of words that did not neatly fit; They being grammatically plural and he also having a specifically masculine use that wasn't clearly distinguished from its generic use upset the same sort of people that got upset by less being used of both countable and uncountable terms while fewer could only be used of countable.
This encouraged people to desire such a pronoun, but it wasn't a pressing need.
Later, the undesirable ability of generic he to carry an assumption of masculinity with it made it increasingly unwelcome, especially when the political implications gained more attention, such that it is now largely obsolete, but they and "he/she" still served enough that most people didn't feel any need to adopt ze, thon,* or other suggested pronouns.
In particular, if someone wanted to specifically indicate singularity and also wanted to be neutral as to gender, but were not considering genders other than male and female (which tended not to be considered until relatively recently) then "he/she" filled that gap. Meanwhile they continued in informal use in almost all forms of English, as much as it was considered improper by some.
Conversely, while people are beginning to be more respectful of other genders, the argument against grammatical plurality being used with singular antecedents has lost favour. As such a reason why people may have favoured he over they 150 years ago or "he/she" over they 50 years ago is no longer as strongly with us.
That they is the pronoun preferred by a great many non-binary and gender-queer people today continues to reduce the pressure to do otherwise. While those who do favour an explicitly singular gender-neutral pronoun will lead us to use ze of zir, or thon of thon, it doesn't lead to a strong pressure to replace they and them in other uses.
In all, while we've had different reasons to be unhappy with the position of they, he and "he/she" over the centuries, we haven't had any reason to be happy with any particular alternative.
*Note also that thon is a variant of yon in some dialects, or of meaning something yet further away than something yon, which wouldn't have been a big pressure against adoption, but would still have been a slight pressure. There are of course plenty of homonyms in the language, but being a homonym does add resistance to adoption of new terms.