I don't understand, because even though Mike hasn't gone to Harvard, it is expressed that Jessica can waive the rule as she did with Rachel going to Columbia.
In many states, including New York, it is required that a person obtain a law degree before being allowed to take the BAR exam. Mike has only ever taken the exam for other people. Then, in order to be a representative in legal proceedings, a person must have passed the BAR.
Additionally, there is the prestige that the firm receives from only employing Harvard graduates.
I think it matters more that he does not have a formal law degree than not going to Harvard. He cannot practice law without a formal law degree and that can get him into deep trouble.