So if someone said "what are they doing?" about someone some distance down the street, you would find that objectionable? I can't really think of a less awkward phrasing. "What is that person doing?" "What is he or she doing?" Those both seem like unnatural ways to phrase the question. What would you say in this scenario?

You may have heard this from Culture War-adjacent content. In reality, this is wrong, and you would struggle to find a real employed academic who agrees with you.

Also, languages aren't dictated by a standards body like C++ is. The rules are written to capture the way the language is actually spoken. If the rules diverge from the language, the rules are actually the thing that is incorrect! This is the difference between "de jure" and "de facto" rules.

Buddy I was literally taught that in my (British) English class, and later seen it used in this context by natives countless times. If anything, using "he/she" is an ugly unpronounceable abomination.