I mean, using a web app for the first time would be no different then installing a mobile app - I wouldn't be surprised if I had to give it a few permissions.
[1] https://developer.mozilla.org/en-US/docs/Web/API/Permissions...
It would only work if many users have disabled exactly the same APIs as you and all other non-disabled APIs don't provide any information useful for fingerprinting.
If I could just feed it random data instead of fully disabling, that would also be fine.
http://plnkr.co/edit/hKvPpEZQGCSkCNKly5Z3?p=preview
It would need to remember your choice so you don't get prompted everytime, but that's a small/easy change to make.
It might be a fun project to start though. I've been really enjoying testdouble's API (and have started using that for my unit tests).