Bad grammar

make sure they are pristine. Do you hear? Pristine!

In this case you use a term 'case' which doesn't apply. What you're looking for is 'number'. English number is a category if two members; singular and plural. Some words fall outside the category, they are mass nouns and do not follow the rules that countables do.

Another problem is that the sentence you criticize is perfectly ok.
The word "they" is a perfectly good anaphoric pronoun to use for an unknown entity.
"Someone called - they didn't leave a number"
"Someone has been in here - they did a good job of cleaning their tracks"

In a way the unknown entity (and the party it may represent) takes on the properties of a compound mass noun, compare: "People [b]are[/b] moving in the treeline"
"Someone from Alpha called, they [b]are[/b] ready at the treeline"

All it really means is that focus is shifted from the actual person calling to the potential party that person represented in calling.
This ties in with communicative intent; if we wish to continue talking about the person calling, then we can choose "he" or "she" - but if we wish to continue talking about the client (the economic entity represented by the person calling, even if it's a one-person company (or even an abstract Mr.-Johnson-as-a-client-not-as-my-hairdresser-or-life-long-friend), then we can choose "they" - to foreshadow this shift in topic and so help our conversation partner make the transition.

While 'he' is also the neutral gender and serves equally well.

And where you want to lob charges of political correctness, your targets generally mean to be inclusive, not to be "pc". Calling this "political correctness" is already an insult created by people who somehow fear things like not using 'he' as the neutral gender. So "political stupidity" is superfluous, but could probably fit most of the people who make these complaints about the grammar while attributing this society-destroying "pc" motive to the writer or speaker, whether or not they are also being pedantic. But most of the time there is no pedantic argument at all, and the reasons the usage is supposedly wrong (number agreement, only 'he' is correct for singular neutral) aren't even used. On the other hand, I've never ever seen anyone lectured by even the most truly ardent "pc" nut about using using something other than 'he' for singular gender-neutral pronoun. I think the assumption about political nature of the people who are politicizing pronouns needs to be re-examined.

The pedantic arguments are made endlessly elsewhere.

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