As PP have said, sometimes the use of they/them as a singular sounds perfectly natural. At other times it sounds odd and will be jarring to a native English speaker.
Yes, it has been used for yonks to describe a non-specific person, but depending on the way a sentence is constructed, at times it can sound awkward and unnatural.
And now of course, non-binary folk have claimed they/them pronouns so it’s quite a politically loaded subject.
I had a conversation with a fellow parent the other day and she was describing a non-binary child and problems they were having with school. All fine until we got to the part about the timetable not being available. It was incomprehensible whether “they” referred to the child, the teachers, or the school not having it. The conversation was painful to follow, even though it had started out easily enough with the use of they/them pronouns.
I’m really not interested in the sex of other people’s babies so not knowing wouldn’t be an issue for me. But depending on how you used they/them in a sentence, I think it could potentially sound odd and a bit pretentious (sorry!)
There’s loads of generic ways to reference your baby without using they/them in an unnatural way. I also think there are plenty of ways to use they/them without it sounding contrived or forced.