It's the exact opposite. Without User-Agents, sites need to depend on feature detection, and closing the feature discrepancy between Chrome and other browsers is more complicated than just spoofing your UA to get Google to serve you functioning versions of their products.
1. Our Chrome detection relies on the User-Agent header.
2. But people can just lie in the User-Agent header.
3. Let's get rid of it and use something that's harder to lie about.
Closing any feature discrepancy isn't a goal here, as far as I can see. The whole point is to lie to the user that a feature discrepancy exists when it doesn't.
You can make the argument that Google is free to do their browser detection however they want (and therefore doesn't need to solve this problem by eliminating User-Agents), but this is still an obvious example of the User-Agent header causing problems for Google.
I interpreted your parent's comment differently; namely, if Google's developers can't do User-Agent detection, then internally even they will have to improve how they develop (eg. via feature detection), making their products more compatible with other browsers.
Many people assume Google, as an upper-level business decision, purposely makes products work better on Chrome in order to vendor-lock users to the browser. Maybe that's true; or maybe it's developers being lazy and using User-Agent detection. Removing their ability to do so might actually improve cross-browser compatibility of Google products.
