I don't think I would have even realized why I felt tension reading if you hadn't mentioned this. They/their wasn't confusing at all but, giving the hypothetical user a name was the weird part. I realize now I was expecting some other user to enter the scenario the whole time. Alice and Bob style. When I got to the end, I felt like I missed something. If there's just one, "the user"/"they"/"their" is fine.