This is a good idea, and is something I've thought of for a while; the user agent header was a mistake from both a privacy and a UX perspective. Ideally, web browsers should attempt to treat the content the same no matter what device you are on. There shouldn't be an iOS-web, and a Chrome-web, and a Firefox-web, and an Edge-web; there should just be the web. In which case, a user-agent string that contains the browse…
I can understand why this is a good thing for privacy. Like many things to do with security on the web though, it's just a shame that bad actors have to ruin so many things for legitimate uses. (The recent story on Safari local storage being another example of that...)