I wonder why you feel that singular they is informal? There's nothing wrong with the construction. It's attested as far back as 14th century. And, as you mentio

I wonder why you feel that singular they is informal? There's nothing wrong with the construction. It's attested as far back as 14th century. And, as you mention, it certainly fills a void in the language. https://en.wikipedia.org/wiki/Singular_they https://en.wikipedia.org/wiki/Singular_they