Hole h in conventional theory c exists, therefore c is invalid and alternative theory a is correct is the kind of illogical reasoning that is the underpinning of so many conspiracy theories. Alternative theories need to have their own support for them, not simply rely on missing gaps of knowledge in the conventional theories.
When an argument hinges on the sentiment “it seems unimaginable…” it often means that circumstantial evidence is being recast as something more definitive than it is.
Alternately, if they couldn't read they'd have to go see his plays performed. That Shakespeare really was a shrewd businessman.
His children were all female, no? You didn't educate women in those days.
You are applying 21st century first world standards to 16th century denizens.
No, actually. He had one son, Hamnet, who died aged 11.
There are thousands of factors at play but the credibility of the authors of the theory plays a huge role. Most conspiracy theorists are self-learned "experts" not from the field of inquiry but often coming from other scientific disciplines that provide them with the air of some scientific credibility even if they are laymen in the area of inquiry. In fact, that's such a hallmark of conspiracy theories that I would call those without that property "alternative theories" instead.
In case of Shakespeare, many other theories are just alternative theories, I suppose. Anyway, unless you're an expert yourself the credibility of the authors is key.
You can reject the accepted theory (the standard explanation) if you can falsify some part of it -- this is literally part of the scientific method. Alternate theories arise because the accepted theory has serious holes in it. Some of these alternate theories turn out to be circumstantial, some turn out flimsy and silly, and lots turn out to be ludicrous conspiracy theories. Reasonable people who hear the wild conspiracy theories often wrongly conclude that the standard explanation must be right because the alternate theories are so wacky.
To use you own example: The standard Shakespeare, the one from Stratford-upon-Avon, allowed his children to grow up illiterate (this is an agreed fact among Shakespeare scholars). For someone who valued the English language so much, who was one of the greatest writers of the English language, it seems unimaginable that he would not have made any effort to see that his children learn to read and write. For me, that alone is enough to reject the standard explanation (though I can cite other holes). It doesn't mean I can prove who the real Shakespeare was. It certainly doesn't mean that I accept the crackpot theories.
My point is that if the accepted theory has serious flaws, and you reject it, that doesn't mean that you have to choose one of the crazy theories, or circumstantial theories, or any theory. It's enough to say that the standard explanation is wrong or deficient.