I never understood why someone would want a blue check mark that verifies him as the owner of an account. That verification means that Twitter can prove that you own that account, because they have a copy of your ID, passport or driver's license. So you are legally responsible for all of your content, while people with an unverified account can always say "That is not really me!". I am not just talking about serious crimes, but also simple copyright violations. Posting a photo for example without approval of the copyright holder can get you into serious legal trouble.
And even if you do not post anything controversial, you might not want all your tweets to be connected to your real life person. If I apply for a new job, I do not want my potential employer to read all my tweets. That could cause a kind of self censorship in your head. Should you really tweet that you like good red wine or may people think that you are an alcoholic?
The blue checkmark people are also very often the ones that I am not interested in anyway. So I would love to have an option to block all people with a blue checkmark at once.