Even adopters of it will admit it is trivial to construct a sentence in which it is confusing to determine to whom the pronoum "they" refers.
Alice was having fun with Julia and John, but they were late to an appointment and couldn't stay for supper.
English is already full of omnibus words. We don't want to transform a pronoun in one.
I have never, ever found a single person (in the Internet or not) who took offense or complained when someone mistakingly picked {he,she} to refer to that person when the latter found {she,he} to be the "right" one. That's because:
It is simply an intelligent decision. Use your brain power to do things that actually matter rather than alternating between two pronouns to be "fair".
Women's rights obviously ought to be equal to men's. This is simply logical, no one in his right mind will dispute it.
You don't do that by fighting over pronouns though, but by supporting and pressing for laws, regulations and behaviors that promote such rights.
Using "they" when the gender of the subject cannot be ascertained benefits no one.
Don't go for red herrings.
A related, awesomely enlightening reading related to the subject: Is the term "man hours" appropriate for the work place?.