Actually he wasn’t all that well known as a writer in his lifetime, and when he died his will contained no references to any manuscripts, books, a desk, or writing materials. His daughter was surprised to hear he was thought of as a writer. He was not the rockstar that someone like Kit Marlowe was.
There are some legitimate reasons to question the authorship of the Shakespeare Canon. One doesn’t have to be a crank to do so; scholars have been wondering about it for a long time (particularly since the renewal of interest in the Canon a few hundred years ago).
The family of the great jazz innovator/pianist Jelly Roll Morton were surprised to learn about his importance — and this was in the 1950s. Families often don’t have the slightest idea of what great artists are doing, especially if they don’t live in the same household.