The opposition listed here is mostly misguided. For example, there is ~0% chance Google breaks "screen readers" or "assistive technologies" with Web Integrity. Saying that they would break seems like a bad faith argument to me.
Fundamentally this is about DRM and whether or not you think it should be allowed. I believe optional DRM for web applications would be good for developers and users of the apps. It would be bad for most other entities (crawlers, scrapers, criminals, etc). So I am in favor of it.
One option would potentially be to only allow positive assertions - ie: "You are up to date" not "you are not running this". TBH I think that would address a major concern for me.