You have fallen into the trap of the fallacy of negative proof.

Fallacy of negative proof: "an attempt to sustain a factual proposition merely by negative evidence"

The belief that Shakespeare did not exist because of a lack of documentation, or the lack of a birth certificate falls into this fallacy. The negative proof of no documentation does not dismiss the fact that there is documentation of Shakespeare's existence.

When specifically talking about the lack of a birth certificate, it is important to remember that we cannot impose our ways of proving someone existed on another time period. This is especially true since it was not common practice in those times to have a birth certificate. The baptismal record is as close to being an equivalent to the birth certificate of the time period. Ergo - as far as historians are concerned, Shakespeare existed.