English teacher cosign. Though I do usually advise students to be aware of their audience, and avoid the singular "they" if it will distract pedantic readers. (Essentially, and unfortunately, you have to advance to the point in life where people are no longer searching for reasons to use a red pen.)

Stop! Abomination! Monstrosity! <g>

I tells you whats: you convince Jane Austen to revise and correct Pride and Prejudice, removing the singular "they", and I'll concede the point.

I'll make you a deal, Thomas: If you'll lobby equally vigorously for Jane's use of comma splices [1] [2] as acceptable practice, I'll reconsider the singular "they."

[1] E.g., the second comma in "It is a truth universally acknowledged, that a single man in possession of a good fortune, must be in want of a wife." (Chapter 1, first sentence.) Or the third [EDIT: fourth] comma in "Not all that Mrs. Bennet, however, with the assistance of her five daughters, could ask on the subject, was sufficient to draw from her husband any satisfactory description of Mr. Bingley." (Chapter 3, first sentence.) [3]

[2] https://en.wikipedia.org/wiki/Comma_splice

[3] https://www.gutenberg.org/files/1342/1342-h/1342-h.htm#link2...

Those aren't comma splices because there aren't two independent clauses.

True -- is there a better term for the comma usage illustrated in the two Jane Austen quotes?

Superfluous, stylistic commas. Or superfluous stylistic commas if you prefer. Commas are inserted seemingly at random by some authors to control the pacing of a sentence. It's not necessarily wrong. btw, "confessions of a comma queen" in the new yorker may be an interesting read.