Never met a writer who wasn't a voracious reader. "Met" largely refers to interviews where writers really open up and give us more insight into their personal lives that we expect. I invariably learn that they are big readers and most, if not all, have their favourite writers. Also many say they are inspired by certain writers - for some it's reflected in their writing, whilst for others you'd never guess the source of their inspiration. Regardless though, I think writers just write because they want to and because they can and reading is also an important part of their day.
Shakespeare works for me mainly through movie adaptations. I stopped reading his works when I left school. He may be an inspiration for plot but I would not recommend him for language
Thou may hath trouble with thine audience.