1. Unlike EME (the controversial web DRM backed by Google that was standardized somewhat recently), the Web Integrity API requires a third-party service, which involves maintenance costs, as well as development costs to constantly adjust to the arms race against all the hackers who really want to thwart these tests.
2. In a "functioning attestation industry", many attestation servers would compete on price to validate users, making the network efficient and robust. I struggle to see this becoming reality because decent attestation would require very complicated techniques for each supported browser, and there is only 1 company that does both significant browser development and also wants to run an attestation server.
3. In a monopolized attestation industry, Google would be the single point of failure for all DRM-protected media on the internet. Google's down? So is Netflix, Hulu, HBO, etc. because they can no longer validate that their users are running an approved version of Chrome. This also give Google an incredible amount of leverage over other companies, because they can change fees and policies unilaterally and there are no alternative games in town. Companies have an incentive not to put themselves in that position.
If the entire media industry coalesces around Google Chrome as the only supported browser for media on the internet, and bestowed this incredible market power and leverage upon Google, then it could work. I find it hard to believe that this will slip past every significant regulatory body on Earth, and any significant gaps in market control would make the scheme unworkable.