Of course this circular argument goes both ways.

One way you start with the assumption that Shakespeare didn't have whatever expertise was involved to enable him to write the plays, and therefore he couldn't have written the plays.

Going the other way you start with the assumption - consistent with contemporary testimony - that he wrote the plays, from which follows the conclusion that he must have had whatever expertise was involved to enable him to write them.

In fact, even it it were demonstrated that a high degree of formal academic education would have been required, which is questionable, there is no particular difficulty in putting together scenarios in which he could well have acquired this, and which also suggest why he might have wished to keep a low profile in some respects. "viz)