How is it possible there is still lingering doubt about whether Shakespeare wrote all the works attributed to him, given he existed in a relatively modern period of our history?

There isn't lingering doubt in the scholarly community that Shakespeare wrote his plays. However, the history of the authorship conspiracy theories is pretty interesting in itself. See this previous thread by u/Texpeare for details on how we know Shakespeare wrote Shakespeare and this thread u/Harmonia for the history of the conspiracy theory.