For Knuth's sake: The GDPR is NOT about cookies! The older 'cookie directive' is also NOT about cookies! They're about a third party storing their data on your computer, or storing your personal data on their computers - no matter what technology is used.
Nothing in the GDPR stops websites from honoring "Do not track" and then _not asking_ if it's present. They don't have to ask if they don't track you! They don't have to ask for a technically necessary session cookie that appears after you actively log in!
Websites ask because they want to track you! A 'law targeting browsers' would not help because people would say no to cookies, and then websites would ask about some other way to track you. Because they want to track you.