Permanently unfollowing all verified users wouldn't even remove all the verified tweets from your timeline, since non-verified users might retweet verified users.
Turns out something that does this already exists!
The creator of https://github.com/insin/tweak-new-twitter dropped by and brought this tool to my attention.
It's everything I wanted, works perfectly, plugins for Chrome and Firefox.
Check it out! Might be exactly what you need.