Having thought about it for a bit, it's possible for this proposal to be abused by authoritarian governments.
Suppose a government—say, Wadiya—mandated that all websites allowed on the Wadiyan Internet must ensure that visitors are using a list of verified browsers. This list is provided by the Wadiyan government, and includes: Wadiya On-Line, Wadiya Explorer, and WadiyaScape Navigator. All three of those browsers are developed in cooperation with the Wadiyan government.
Each of those browsers also happen to send a list of visited URLs to a Wadiyan government agency, and routinely scan the hard drive for material deemed "anti-social."
Because the attestations are cryptographically verified, citizens would not be able to fake the browser environment. They couldn't just download Firefox and install an extension to pretend to be Wadiya Explorer; they would actually have to install the spyware browser to be able to browse websites available on the Wadiyan Internet.