If you are the client receiving letters from professionals such as solicitors, Doctors, MPs, accountants etc, you will find they sign off first name, surname. They do because to use ones title in signing off is to imply superiority compared to the person being written to. It's why teachers shouldn't sign off with Mrs X. It's nothing to do with kids not knowing teachers names or parents not knowing them. It is a professional convention, not one related specifically to schools.
Most teachers don't have a problem in meetings in introducing themselves as 'hello, I'm Jane Smith, Toms' French teacher'
However,mig then would be correct for parents to refer to Mrs Smith, not Jane or Jane Smith in both writing and in person. The teacher would also refer to the parent as Mr/Mrs.
It's not bout asserting authority, because neither person refers to themselves as Mr/Mrs etc, ONLY to the other person as that as a sign of respect - it is used by both parties.
It may well be that in some industries when clients are involved things are more informal. Often these are more regular contacts, whereas teachers in secondary schools may only see parents once a year and never or only extremely occasionally write to them. There is a distance between teachers and parents which means the relationship remains formal and Mr/Mrs isn't being overly formal, but appropriate.
Teachers who sign off as Mr X or Mrs X are mistaken in how to write. They are muddling how they would refer to a child and how they would refer to a parent, who is an equal. They should sign off as first name, surname and out a title in brackets if needed - and it is helpful, so people know the title to refer to.