I don't think you teach badly; I don't think Shakespeare is badly taught on the whole. (And god knows I would cringe away from being called an 'intellectual').
But I am a working teacher too. And the way Shakespeare (and other bits of English Lit) are taught at school can sometimes do a disservice to students who want to study those things at university. I don't think this is the fault of teachers. I think it is the fault of a curriculum that has been written by people (like Gove) who have arrogantly decided that they can be specialists without specialist knowledge, and can tell what is important.
It really bugs me, and worries me, that this thread has give us the message, over and over and over, that what's important about Shakespeare is 'his language'. And we should learn 'his contribution to the language'. Either that, or we should read the plays as an opportunity to learn good morals: 'this is racist and Shakespeare shows us how' or 'this is sexist and look how Shakespeare exposes it!'
I get why Gove would want to push a myth of 'English Shakespeare' who 'gave us our language and showed how not to be racist/sexist'. After all, if we're learning lessons about sexism from someone who's been dead four centuries, we're going to feel pretty good about ourselves by contrast. And if we really believe that Shakespeare invented English, then we clearly don't need to acknowledge centuries of European influence and exchange, let alone any debts to the rest of the world.
It's a really disturbing trend. And it's that which makes me uneasy, not any individual teacher or the abilities of people who teach secondary school.