While your suggested edits are fine, I can't agree with the assertion I've quoted. The plural pronoun they is very widely used to represent a single person in American English. It is true that some editors and holdout sticklers object to this usage.
Yes, absolutely. As I always agree and often mention here,
they is widely used. Is it universally accepted yet? Maybe. Am I the only person in the English speaking world who thinks plural pronouns only go with plural nouns? I don't think I'm a holdout stickler any more than I think you're lazy, sloppy, or ignorant. I'm not. You're not.
There's also the widely held belief that there are no rules of grammar and spelling, only practices. I agree with that, too. So which practices should we use? If I have several friends who are jazz musicians. All of them are fluent in reading sheet music and have in-depth understanding of music theory. How did they get there? They all say the same thing: First you learn the book. Then you throw it away.
We all get that, right? But what book are we talking about when we talk about learning English?
To put it another way: Where I live, rural New England, not only is
they fine, but "Me and him seen this guy drive their skidder right into the lake" is perfectly acceptable. So when a student writes that, what would you correct and what would you leave alone? Where would you draw the line? HOW would you draw the line.
Any line you draw is arbitrary and subjective. It will probably betray huge class and regional differences, too. The lines we all draw certainly don't have to match. That would be boring! The one I use comes from my own experience. When I studied and worked at the state university, the entire student body and faculty of all the campuses were expected to use the same reference, Dianne Hacker's grammar guide. My friends on the English department faculty and the ESL students who used the tutoring program were all expected to abide by it.
I say "they" instead of "she" sometimes, too, and will happily welcome the day when, like "you," "they" is accepted as a singular pronoun in all formal printed references. (Although there must be some utility to separating singular from plural forms, or we wouldn't have so many regions that use "you all," "y'all," "youse," and "yiz.")
But for the purpose of teaching correct, natural American English, it's short-changing students to let them throw the book away before learning it. (Or, here in central Maine, even opening it!) Today, "Me and him seen" is natural for some people but not okay for all people. When Ms. Hacker says it's fine, I will, too.