It’s actually really tough to figure out how Shakespeare’s plays were viewed during his lifetime. They were definitely considered excellent, though. I’d also argue that “classics” can only appear over time, and that even works that are widely hailed in their own time don’t always stand the test of time. At any rate, Shakespeare’s plays were extremely popular and respected and his poetry was certainly considered to be of very high quality during his lifetime.
As for the Puritans, well, that’s what they did.