1...I think that the first rule for rewriting sentences is that you cannot change the meaning, or even the emphasis. So my first objection to the student's sentence is that it violates that rule. The original sentence at least strongly suggests that the first lesson had started because the student was late. The student's sentence says nothing about 'being late', and we can only guess that that might be the reason the class had already started.
2...I don't like the negative form of 'As is often the case...'; it's awkward, at the least.