The security engineer in me loved this proposal, the privacy/FOSS enthusiast in me hated it.
Security is about making sure that something happen via a certain path, without things going wrong in ways that can be abused.
Web Integrity effectively enables a way for, say, a bank to validate that the request isn't coming from botnet VMs spun up in the cloud, but rather an actual device with an actual human behind it. In fact, many websites already try to accomplish this with fingerprinting, which is why certain browsers and VPNs are already effectively blocked from the web.
Yes, it has the effect of limiting your freedom and privacy, but there is a definite security gain here that will lower the likelihood of account compromise.
Security != privacy; and security != freedom. Often there's a tradeoff. Security engineers are employed by their company to increase security - not freedom, and not privacy (unless mandated by regulation), hence the bias you typically see from them.
https://en.wikipedia.org/wiki/Trusted_Computing#Remote_attes...
It would only ensure that the laptop user does not run software that is not approved by Google or similar, a DRM, as Trojans will continue to find ways to disguise themselves and insert themselves into authorized processes.
The company would find difficult to run custom software not covered by such DRM. In time, the Web Environment Integrity API organisation would say that there are too many authorisation requests or similar and demand economic compensation from developers to be included in the authorized filter scans, at time the supposed initial objective is not accomplished, as this DRM does not ensure the laptop is not infected.
A good example is Xbox, for which there is no known custom software, which in turn guarantees the lack of cheating software.
I am quite sure that the client programs need to be cooperating for the WEI to be practical.