I don’t think he’d like his legacy. He really made plays for the people, now his plays are for the erudite, and the level of worship lit departments give him is embracing.
On your point, the English language has changed a LOT since Shakespeare. How we talk bear no resemblance to him at all. But, because we’re all force fed his stuff in high school, it still rings familiar. If we read as much Chaucer in high school as we do Shakespeare it too would be completely intelligible.
He actually has quite a bit of popular appeal. He’s probably the most popular writer in the world.