That moves the onus to the user to distinguish between tracking and non-tracking uses of cookies and local storage. It also contradicts the principle that tracking requires informed consent.
Speeding is illegal. Controlled substances are illegal. Murder is illegal. Embezzlement is illegal. Driving in a school zone while using a mobile device is illegal.
Hindered, but not eliminated. It is not a panacea. As long as people are willing to flaunt laws, laws are not the ultimate solution. The main reason here is that your laws are regional. Someone operating outside the jurisdiction of said law feels not obligation to obey said law, nor feel fear of punishment of said law.
* Let websites do whatever they want with cookies/local storage.
* Let browsers delete them as often as they want.
* Make other kinds of fingerprinting illegal.