At some point it feels like trying to drain the ocean with a cup. Maybe we just need to accept that anyone who really wants to fingerprint you _can_ fingerprint you unless you use a specialist browser.
At that point the solution is fairly obvious, make it legally difficult to use unique fingerprinting and move on (ie stuff like gdpr). People will still do it, but they'll have to balance it with not falling foul of the law and wont be able to abuse it too much.
We wont stop real world facial recognition by all trying to make our faces more similar either, we have to accept it's generally possible to do, but discourage the actual doing of it rather than trying to make it impossible.
(note in both cases, actually preventing it when you have a reason to is totally possible and valid, via specialist browser modes and physical masks respectively)