I think your entire reasoning is in reverse. Using "she" where no gender needs to be specified is the awkward thing to do, and it's not neutral, it's explicitly designating a female gender. Using "they/them" is far less awkward, the reader doesn't have to have any kind of opinion about the text based on gender associations, making it less awkward.
Occasionally, they/them can get more confusing, especially in a story involving multiple people. It can become unclear if you're referring to just one of the people using singular they, or if you are in fact returning to more people using plural they.
