That's kind of the point of all this. They force websites to enact the verification because they have leverage over businesses that they don't have over citizens, and then they expect that the citizens will hate it so much that they don't go to the "bad" sites at all. "Thank you for your cooperation!"

ETA: (accidental submit; sorry) I'm in the same boat! Not entering my ID information into any website, much less ones they've got on the list. And so they've successfully boxed us in. At least for me, I intend to raise hell about it aside from just not sharing PIA, but I don't have any delusions of it's effect.