>how do people feel about the expanding use of they/them/their as an indefinite 3rd person singular pronoun

I certainly use this form in writing. I have written some "how to" books and avoid at all costs using his/her. I feel very comfortable with your example - "Each student should check their work carefully and correct their own mistakes". Some people would accuse it of being impersonal but I think that it is neutral and the rest of the writing can carry the responsibilty for sounding friendly.