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 browser and even the OS only encourages differences between browsers. Adding differences to your browser engine shouldn't be considered safe.
Beyond that, the user agent is often a lie to trick servers into not discriminating against certain browsers or OSes. Enough variability is added to the user-agent string that a server can't reliably discriminate, but it still remains useful for some purposes in JavaScript and as a fingerprint for tracking.
Which brings me to privacy. It's not as if there aren't other ways to try and fingerprint a browser, but the user agent is a big mistake for privacy. It'd be one thing if the user-agent just said "Safari" or "Firefox", but there's a lot more information in it beyond that.
If the web should be the same web everywhere, then the privacy trade-off doesn't make much sense.