Shakespeare’s achievements, going from a glover’s son to Britain’s most prominent playwright, would have been a cause for anger and jealousy. His rise was a slap in the face to those better off with access to more elite education and more opportunities. It is no surprise, therefore, that his genius would have put him in the crosshairs of his contemporaries. This could explain Greene's malicious missive.
There are also several rivals who concede Shakespeare’s genius and leave glowing reviews. Ben Jonson, Shakespeare’s great literary rival coined the term the “Swan of the Avon”. He references the great playwright’s talent, then places him as being from Stratford-upon-Avon. It is inconceivable that he would do that to his rival if Shakespeare was merely a front for another author.