First of all, look, when somebody says, "Everyone needs to turn in their homework." it doesn't bother me too much. I still use "his or her"; but, most people use "their" in that situation because it's easier and still gets the point across.
The whole "Can you ask them when they're coming over?" part is playing hide the ball. If you're having multiple people over, of course you would use plural pronouns. If you're only having one person over, you use singular pronouns. A normal person wouldn't say, "Can you call Julie and ask them when they're coming over."
If somebody were in an accident and in a coma for the last several years, woke up yesterday, and were still familiarizing him or herself with the modern world and that person heard another person say, "I saw Demi in concert last night and got to meet them after the show." it would sound like "Demi" is an band and not one person.
Still, there's a bigger issue of principle here. Language is important and these people are trying to manipulate it from the top down rather than treat it for what it is -- an emergent order. There's a huge overlap between these people and the people who constantly use the word "woman" twice in a sentence while refusing to define the word. They're trying to put words into our mouths. That's downright Orwellian.