What if the school asks the students to submit the letters themselves?
I had this experience as a letter writer (in France). I just sent my letter to the applicant and she uploaded it. My letter was very positive so I did not think much about the situation although it did seem weird to me.
I think in some US Universities (in California?), when you submit your letter, you should indicate whether you allow the applicant to look at it or not. I do not write bad letters so I always indicate that I do not mind.