The only issue I have with ‘they’ and ‘their’ is that I was taught in my English classes that this doesn’t reference who ‘they’ and ‘their’ are and that a person might ask a question as to whom these statements refer. In other words; “Who’s she, the Cat’s mother?”
I have been experimenting with neutral-gender antecedents as I am a science fiction writer who likes to explore the less common types of races there could be. In the past I was experimenting with a race that had 4 sexes; male, female, baby-carrier and dronal. You can’t refer to the dronal or baby-carrier as ‘he’ or ‘she’ because they don’t fit the part (and don’t have the parts).
I notice (but bear in mind that this is a generalisation and there are exceptions) that a large swathe of African and Asian bloggers, especially males, have difficulty connecting with a wider audience because they only reference male antecedents. I feel they are missing out on a much bigger market by being so narrow in their focus.