It’s true that the magnitude of Shakespeare’s genius was not fully recognized during and shortly after his life, but there were leading indicators. The fact that his reputation is greater 400 years after his death, than 40 years after his death shows is that genius shows in the test of time, not the fads of the day.
He was recognized as the greatest of his generation back then, but he both died young and the era of the Elizabethan / Jacobian live stage didn’t last all that long. During Victorian times Shakespeare was revived, but the scripts were doctored to remove some of the bawdiness, add then-current euphemisms, and to add stupidities like, Romeo and Juliet survived.
The 37 known surviving full plays are not all equally accessible today, for sure, but appreciation for Shakespeare really has picked up in the 20th century, and movie treatments began early in the silent era (I think the first one was a one-reel production of “King John”). In the 1970s the BBC produced all 37 plays, and a year or so ago I read somewhere (maybe here on FR, if so, probably in a topic you posted) that the BBC was preparing to do so again.
I see a live stage production of one Shakespearean play once a year, and am sometimes surprised at how things are done. A few years back I went to see “Romeo and Juliet” — easily one of the best known plays in history — and wasn’t expecting to be knocked on my ass. The only problem with the play is that inquest at the end, it reminds me of the Bob Cummings role in “Dial M for Murder”. :’)