There were rumors during Shakespeare's life
His contemporaries at the time appear to have harboured suspicions.
Unlike his contemporaries, there is no historical documentation to suggest Shakespeare made a living from writing plays.
Other writers at the time left documented evidence that they earned their living from writing. Even in cases where the writing itself has not survived, there are letters and payment records that showed they earned an income from writing.
There is almost no indication in historical documents that Shakespeare ever earned an income from writing, despite his life being exceptionally well documented compared to most in Elizabethan England. We know he was an actor, had a stake in a theatre company, was a moneylender, a property investor, a tax dodger and had a restraining order imposed upon him. But nowhere do we have historical documentation pointing to an income earned from writing.
The fact that there is more documentation about William Shakespeare than any other figure in Elizabethan theatre shows that he had the fame and public prestige we would expect from someone who wrote the plays. Absence of evidence is not evidence of absence. Just because there is no surviving documentation to show that he made his livelihood from writing does not mean that he didn't. It is possible there are just no surviving documents. There is extensive evidence which places him in the theatre world. He was in the Lord Chamberlain’s Men, a troupe which would go on to become the King’s Men. But, perhaps most convincingly, his name is on the plays. Why would there need to be any further documentation beyond that?
[P1] There is no documentation showing that Shakespeare earned any money from writing. [P2] There are many documents that show his contemporaries made their livelihoods from writing. [P3] Therefore, Shakespeare did not make money from writing. [P4] Therefore, he can't have written the plays.
[Rejecting P3] Absence of evidence is not evidence of absence.